Is there a more "physically mature" way to think about the right hand rule with electromagnetism? I've always found using the righthand rule to remember how forces, B-fields, and particle velocities to be intellectually cheating myself a bit. It feels like being able to multiply numbers by using your fingers without knowing what multiplication really is. Yet, when I try the "just think about it approach" I still end up just imagining what my hands would do in my head. How can I think about the directions of the electromagnetic right hand rule vectors to intuitively understand what's going on, rather than using the equivalent of a cheat sheet?
 A: This post is partly a direct answer to the question, partly a response to some issues raised by the other answers, and partly a soapbox issue for me that spins my urge to write out of control, oops.
The question becomes somewhat more interesting if you look beyond electromagnetism.  Within electromagnetism there is no difference between using (consistently, for all cross products) a left-rule instead of the conventional right-hand rule.  Outside of electromagnetism, we do find axial-vector quantities which are physically observable. However there is still an arbitrary choice as to what label is appropriate for those observable axial vectors.
 1. Electromagnetism is ambidextrous 
As your existing answers say: the direction of the magnetic field $\vec B$ is not actually an observable.  What we observe in experiments within electromagnetism are either vector quantities (forces, accelerations, displacements) or scalar quantities (energies, masses, interaction probabilities).  Any observable in electromagnetism has used the right-hand rule an even number of times, which is indistinguishable from having used the left-hand rule an even number of times.
For example, consider the Lorentz force on a moving charge or a current:
$$
\vec F = q\vec v\times\vec B = I\ \mathrm d\vec L\times\vec B
$$
Here the second cross product is hidden in the orientation of the magnetic field, which has been produced by some other current $I_2$:
$$
\vec B = \frac{\mu_0}{4\pi} \int \frac{
I_2\ \mathrm d\vec\ell \times \hat r
 }{r^2}
$$
You can describe the Lorentz force as a current-current interaction, with the following hands-free rules:

*

*parallel currents tend to attract

*antiparallel currents tend to repel

*skew currents tend to feel an aligning torque

You should probably be able to convince yourself that a big floppy shoelace, carrying a current, would minimize its magnetic energy by stiffening into a circle.  Furthermore, two such circular current loops will minimize their energy by reorienting so that the nearest bits of each loop are the most parallel to each other, which is the parallel-dipole attraction which we teach in kindergarten using permanent magnets.
While this current-current formulation may satisfy your desire for an ambidextrous mnemonic, I find it much more challenging to ask "to which distant current will this moving charge tend to parallelize" than to ask "how does the trajectory of this moving charge evolve in this region of magnetic field."  The field treatment makes it much simpler to keep track of whether the contributions of near currents and far currents cancel exactly or not.  The field treatment has the advantage of being local.  Furthermore, in quantum mechanics we find magnetic interactions between particles which are intrinsic magnetic dipoles, and insisting on a current-current model would require us to invent fictional microscopic currents with some unpleasant properties.
 2. Describing the weak interaction still requires a reference artifact 
Mathematically, we describe the ambidextrousness of electromagnetism by defining a "parity" or "inversion" or "reflection" operator,
$$
\hat P f(x,y,z) = f(-x, -y, -z)
$$
If the person next to you at the bathroom sink is holding their toothbrush in their right hand, their reflection is holding its reflected toothbrush in its left hand.  Beware that a plane mirror only switches "towards" versus "away," $f \to f(x,y,-z)$, which is an inversion followed by a rotation.  It's also worth noting that inverting twice is the same as doing nothing; that is, $\hat P(\hat P f) = f$ gives the same result as the identity operator.  When we say that "electromagnetism is invariant under parity transformations," we mean that all of the mathematical terms in the energy of an electromagnetic system are scalars.
The weak nuclear interaction is special because it includes pseudoscalar observables.  The best-known is the asymmetry first observed by Wu and her collaborators, in which a population of radioactive cobalt nuclei were polarized so that their intrinsic spins mostly pointed the same way.  Wu observed that the decay electrons were more likely to be emitted in the direction of the nuclear "south pole" than in the direction of the "north pole": a correlation between the nuclear spin $\vec\sigma$, an axial vector, and the electron momentum $\vec p$, a polar vector.  This correlation is required because the decay products must carry away one unit of spin from the parent cobalt nucleus, so the half-units of spin carried by the electron and antineutrino must be parallel.  However, the "weak charged current" couples only to left-handed particles and right-handed antiparticles: the antineutrino and the beta electron emitted in the decay must generally have $\vec\sigma_{\bar\nu}\cdot\vec p_{\bar\nu} > 0$ and  $\vec\sigma_{e^-}\cdot\vec p_{e^-} < 0$.
While this asymmetry in the weak interaction solves Gardener's Ozma problem of unambiguously distinguishing left from right, it doesn't really eliminate the need for a reference object.  My description of the Wu asymmetry, in terms of "north poles" and "south poles," gets rid of a biological reference that most humans have two hands which are approximately reflections of each other, and that the hand on the same side as the liver tends to be stronger than the hand on the same side as the heart. But the replacement reference is that most humans who study physics have also studied geography — specifically geography in the tradition of Mercator, who lived in Flanders and put his house in the top half of his map.
We could perhaps (as did Gardener, and in the spirit of recent improvements to the SI) choose to define the sign of angular momentum using the preferred polarizations of neutrinos and beta electrons. But neutrino polarization is not terribly helpful as a mnemonic, and the way that I personally keep track of mathematical "north" is to imagine grabbing a globe with my right hand, fingers to the east.
 3. Mathematical definitions still contain an arbitrary phase 
You have a number of answers which suggest mathematical definitions, such as the Levi-Civita symbol.  This doesn't really buy you anything.  Continuing with the weak interaction, consider that the Standard Model Lagrangian is an extension of the Dirac equation, which predicted antimatter using a symmetry argument that the electron field should contain two spin states each with two charge states.  That is, the Dirac electron is a four-component "spinor," with a representation like
$$
\psi = \left(
\begin{array}{c}
\displaystyle{ e_L^- \choose e_R^- } \\
\displaystyle{ e_L^+ \choose e_R^+ }
\end{array} \right)
$$
These spinors are manipulated using the four-by-four gamma matrices, four of which $(\gamma^0,\vec\gamma)$ transform into each other under rotations and boosts in the same way as a relativistic polar vector.  It turns out to be useful to define a fifth matrix,
$$
\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3
$$
which has (among others) the property that $\gamma^5 \cdot\gamma^5$ is the identity matrix.
Its eigenvectors are the left- and right-handed parts of the spinor,
$$
\psi_L = \frac{1 - \gamma^5}2 \psi
\qquad
\psi_R = \frac{1 + \gamma^5}2 \psi
$$
where the "left" spinor is the one with eigenvalue $-1$.
Many discussions of parity violation in the weak interaction, especially older ones, will say that the sign of the parity violation means "the weak interaction is $V$ minus $A$" somewhere near where they say "the weak interaction is left-handed." The "minus" refers to the sign in $\psi_L$,
because a term like $\bar\psi_L W \psi_L$ appears in the Standard Model Lagrangian, while $\bar\psi_R W \psi_R$ is missing.
But: this is all just a convention. Suppose instead that we liked left-handed cross products, and therefore left-handed coordinate systems.  The tool for getting to the mirror world is the inversion operator. Our rules for inversion require that $\hat P \cdot (\gamma^0,\vec\gamma) = (\gamma^0, -\vec\gamma)$, so the effect on the fifth matrix is
$$
\hat P \gamma^5 = (-1)^3 i \gamma^0\gamma^1\gamma^2\gamma^3
$$
We still have the same physical object which is coupled to the $W^\pm$ boson. Perhaps we should have called it $\psi_\text{Larry}$ to emphasize that the $L$ is just a label, because our mirrored person would believe that we've got the eigenvalue wrong, that $\gamma^5 \psi_\text{Larry} = +\psi_\text{Larry}$, and the mirrored history books would say "the weak interaction is $V$ plus $A$."
 4. You do not have telekinesis 
As a philosophical matter, if you can change a number just by thinking about it differently in your brain, that number is not a physical observable.  A comment on another answer muses

It seems like the very nature of 3D space would make you "choose" one [handedness] or the other.

This both is and isn't true. Three-dimensional space isn't complex enough to have an intrinsic handedness.  But what we mean when we talk about "the Standard Model" is really a list of properties exhibited by the vacuum of our four-dimensional spacetime, and our four-dimensional spacetime does have pseudoscalar observables.  Spacetime is "screwy" in the sense that, like a threaded fastener, there is a correlation between angular momentum and linear momentum.  I don't think it's known whether this correlation really is intrinsic, in a way that our primate brains could find both "intuitive" and "necessary."  Another possibility is that this weak intrinsic screwiness is a spontaneously broken symmetry, like the direction of the polarization of a ferromagnet, and if we could get a region of spacetime to be hot enough the symmetry might be temporarily restored and break into a different direction when cooled.
But for electromagnetism, the only role of handedness is in communicating about intermediate results. And as far as "maturity" goes: in the community of researchers on parity violation, everybody will occasionally stop themselves mid-sentence, make a little screwing motion with their right hand, and say "no, wait, I meant this way" before continuing.
A: Other answers establish well that this is an arbitrary convention, but you are also wondering why when you try to think about this convention you always come back to imagining your hand. This is because there isn't a way to communicate chiral directions (left/right, clockwise/anticlockwise) without a physical reference artifact (or possibly a particle accelerator and knowledge of CP violations.)
The fact that we call the rotational directions "clockwise/anticlockwise" is a good hint for this - the best way to talk about it is to just reference a familiar artifact, a clock. If you were talking to a human raised in a featureless box, you would have absolutely no way to tell then which of their hands is their right hand, unless you used the asymmetry of their own body (e.g. telling them their heart is on the side designated "left," and hoping they weren't born with situs inversus!) If you were talking to an alien via radio, you'd be hard pressed to synchronize your understanding of direction with theirs, unless you knew their location and could make a physical artifact out of the stars (e.g. telling them that a certain sequence of pulsars goes "clockwise" around Earth from their perspective.)
So if the right hand rule feels childish, just remember that it's only as childish as the word "clockwise." No matter how wizened you are, you learned chiral directions from an artifact and simply memorized that artifact.
A: To expand on hft's answer: in my opinion, the most "physically mature" way to think about this is to internalize the (rather Zen) understanding that pseudovectors (like torque or magnetic field), which result from cross products, don't have any physically meaningful oriented direction at all.
I used to believe that if we suddenly decided that we'd globally switch all cross products to use the left-hand rule instead of the right-hand rule, then all of our physics formulas would stay the same, except that some formulas would pick up a minus sign to correct for the new orientation of cross products. But this actually isn't correct. In fact, all of our physics formulas could safely stay the same identically.
After making this convention change, all of our pseudovectors would indeed suddenly change direction - but none of them have a directly physically measurable direction anyway. You might say "Nonsense - there are certainly sensors that measure the direction of a magnetic field." But they don't directly measure it (although that claim will probably unleash philosophical debates about the meaning of the phrase "directly measure"). Instead, the sensor internally takes a second cross product of a test charge’s velocity and the magnetic field, and measures the direction of the resulting true (polar) Lorentz force vector. So, as long as you consistently use the left-hand rule, you'd still get the same polar force vector inside your sensor, and only your interpretation of its reading would need to change.
So the way I see it, pseudovectors truly don't "exist" as vectors in physical space. They're only ever placeholders to later be inserted into future cross products that will eventually turn them back into physically measurable polar vectors. So which hand you choose to use is truly irrelevant - you don't even need to change any signs in your formulas - as long as you always use the same one.
A: Locations in Newtonian space can be described as 3 coordinates, or a 3 dimensional space.  So can forces and velocities and the like.
As it turns out, the ways things can spin can also be described by 3 dimensions.
Imagine you have something that is spinning.  First, draw a unit sphere around it, and where the axis of spinning crosses the sphere record those points.  Those two points are on opposite sides of the sphere.
This doesn't fully describe the spin, but tells you where the axis is.
We then need to add on how fast it is spinning, and what direction.
For how fast, we can scale up the distance from the origin.  This is nice, because "no spinning at all" has a distance of 0, and that corresponds to the point where the "axis" is undefined.  Pretty perfect really that they cancel out!
The next problem is what direction the thing is spinning in.  Right now we have 2 points on the sphere and two directions we need to distinguish between.
The right hand rule is literally saying "Ok, we'll pick one of these two points to distinguish spinning one way from the other".  If you selected the other point, all of the math would still work, you'd just have a left hand rule, and some functions for converting between angular vs linear math would have a sign change sometimes.
The point we assign to a spin isn't a velocity or an offset.  It is just something that behaves mathematically like an offset.
The natural mapping between the space off offsets and the space of rotations -- or, the space of vectors and bi vectors in 3 dimensions -- has a choice to make -- there are two equally good mappings with nice properties.  Our choice of which mapping we use is the right hand rule.
To map this over to something you might be more used to, where is the zero point of gravitational potential energy?  Sometimes you end up putting it at infinity, other times at ground level.  Where you put it doesn't matter; but it does matter you do your math while agreeing on where it is.
You can avoid having to make that choice by only talking about differences in gravitational potential energy, but that sometimes makes the math a tiny bit more annoying to use.  So often you end up picking some height or point and saying "all of my gravitational potential energy values are relative to this point".
That choice -- of where 0 is -- is free, but has to be agreed upon in all the math.  The same is true of right hand vs left hand rule -- the choice is free, but it has to be the same or the math gets messed up.
We could avoid making the choice, but then it makes other math more annoying to do.  So we pick one and stick with it.
A: 
How can I think about the directions of the electromagnetic right hand rule vectors to intuitively understand what's going on, rather than using the equivalent of a cheat sheet?

You can't intuitively understand it because it is a convention. We chose the "right-hand" rule and it is now the norm world-wide. We could have gone with the "left-hand" rule, but we didn't.
As I mentioned in the comments, I believe that old (pre-WWII) German physics/chemistry papers often used the left-hand rule (or left-handed coordinate systems).
A: There may be a meaningful alternative to "Right Hand" by relating it to the $xyz$ axes convention and the Levi Civita symbol $\epsilon_{ijk}$, in which if I'm not mistaken, the +1 terms are called "even" permutations of 123, and the -1 terms are called "odd" permutations, e.g. 321.
You could express for example the formula for torque as:
$$\vec \tau = \vec r \times \vec F$$
Or as:
$$\vec \tau = \epsilon_{ijk} \hat e_i r_j F_k $$
This seems to rely on pure math, but I'm not confident enough to assess whether it implicitly assumes a right handedness direction for "123" type axes (xyz).
This is more of a suggestion for further inquiry than then a true answer
P.S. From my pop culture level understanding of QM, it seems that chirality/parity/handedness is an irreducible feature of the 3 Dimensional world that may not have a "deeper" explanation (yet).  The fact that we associate it with our human hands is simply because it is the first manifestation of chirality that people know of.
We could say, "build an xyz axis such that the Kaon decays into a $e^+$ moving in the z direction, and a clock looking from behind (further -z) will advance 'clockwise' from +x to +y  in the direction of the $e^+$'s intrinsic spin...." (cf. Ozma Problem). But this is far more confusing than saying "xyz like your Right Hand" without adding any additional depth.
A: 
I've always found using the righthand rule to remember how forces, B-fields, and particle velocities to be intellectually cheating myself a bit.

The shortest possible answer is as simple as it is trivial:
The right/left hand rule is the most effective summary of the experiment in which moving electrons are deflected by a magnetic field.
The rule describes the behaviour of the electrons by assigning the direction of the thumb, index and middle finger to current I, magnetic field B and Lorentz force F
“I-B-F = thumb-index finger-middle finger"
(the one that must be remembered).

How can I think about the directions of the electromagnetic right hand rule vectors to intuitively understand what's going on, rather than using the equivalent of a cheat sheet?

We only need one rule to be able to qualitatively describe a multitude of processes. Maybe you really just want to learn more about the process of electro-magnetic induction? Otherwise tl;dr.
The convention of the north and south pole
The rule includes two intuitive directions and one convention. The representation of the direction of movement of the electrons (the electric current) through the thumb is intuitive. The representation of the resulting direction of deflection of the electrons by the orientation of the middle finger is also intuitive. But what about the magnetic field? You have to remember that the index finger points to the north pole. Wy we need to remember that?
The magnetic field is a dipole field. It always has two poles, one of which we call the north pole and the other the south pole. There are minerals that are natural magnets or I can magnetise a piece of iron. In both cases, the poles are not marked at first. Both poles of the magnet attract a piece of metal equally. So which pole is the north (south) pole?
There are exactly two ways to assign the poles.

*

*I take an already marked magnet (all of which go back to the one magnet that was first marked with the assignment of the poles (i.e. a pure convention)) and deduce the polarity of my magnet from the repulsive or attractive force.

*I carry out the experiment I-B-F with my magnet and thereby find because of the direction of deflection of the electrons the polarity of my magnet with the right-hand rule. This is possible because the electrons in the magnetic field possess this asymmetrical behaviour. That is their nature.

The behaviour of the other electric charges
The antiproton has the same direction of deflection as the electron. Positron and proton, on the other hand, are deflected in the other direction.
Positron and proton carry a positive charge and the formula of the Lorentz force - which describes quantitatively what the right-hand rule can only do qualitatively - takes this into account. The charge $q$ contains a sign, which reverses the direction of action of the Lorentz force (the deflection of the charges) in its direction.
$\vec {F_\text{B}} = q\vec v \times \vec B$
The right-hand rule also includes the tacit convention that it applies to negative charges.
The Other Electro-Magnetic Induction Processes
You can swap the sequence "I-B-F" cyclically and get the remaining two induction processes - technically used in countless variants.

*

*Already discussed above: The Lorentz force with "I-B->F" describes an electric motor, because a current-carrying conductor in a magnetic field induces a force.

*The causal chain "B-F->I" describes the electric generator, because the displacement of a current-carrying conductor in a magnetic field induces a current.

*The causal chain "F-I->B" describes the creation of a magnetic field, because the forced deflection of charges (as in a coil, in which the charges are forced into circular orbits) induces a magnetic field.

As you can see, the right-hand rule reflect a lot of experimental results.
A: If you want a physically meaningful interpretation: study "Geometric Algebra". It's a Clifford Algebra of space. It's an involved subject that is beyond the scope of this answer.
In short (and I mean short): In this more fundamental, yet unpopular, version of space, cross products are bi-vectors. In 3 dimensions, that concept is represented by a plane (for direction, there is also a magnitude).
So you have a plane: which way does its normal point? Either way: pick a convention. It doesn't change your understanding of the plane at all.
A: The cross product and curl and the like arise, ultimately, from the mathematics of forms, and something called the Hodge dual.
It is too much to completely get into here, but ultimately, in defining these things, you introduce the Levi-Civita symbol $\epsilon_{abcd...}$ where you have the number of indices in the bottom as you have dimensions in your space.  In three dimensions, you end up with:
${\vec A} \times {\vec B} = \epsilon_{abc}A^{a}B^{b}$
and you define the Levi Civita symbol such that:
$$\begin{align}
\epsilon_{xyz} &= 1\\
\epsilon_{abc} &= 0 {\rm \;\;\;if \;any\;of\;the\;a,b\;are\;equal\;(ex. }\epsilon_{xyx}=0{\rm )}\\
\epsilon_{abc} &= -\epsilon{bac} = \epsilon_{bca} = -\epsilon_{cba}
\end{align}$$
This is enough to give you all cross products for general vectors and everything in a "more rigorous way" than pointing your fingers.  The choice of the right hand rule over the left hand rule is choosing $\epsilon_{xyz} = 1$ rather than having it be equal to $-1$.  But it's a lot easier to point fingers than to spend time permuting indices like this.
A: The orthogonal interaction of electrical and magnetic vectors can be described purely mathematically, but when taking theory in to the real world, conventions such as the right-hand rule are necessary.
A: The right hand rule is analog of the imaginary number i and also analog of the right hand rule angular momentum vector. Nature does not follow the right hand rule or imaginary number phase calculation to give a vector pointing perpendicular to a rotating disc or a number opposite of real amplitude. Left hand rule will work perfectly well as well. The more physically mature way to think of it is to know that the artificial rules/numbers are just circuits inside the human calculator. Why humans ought to do it, turning ourselves into calculators, is championed by Renee Des Cartes who pioneered the vector coordinate system.
