In what sense, physically speaking, are electric and magnetic fields perpendicular? I have searched for an answer for in what way are electric and magnetic fields perpendicular, but I only found mathematical explanations speaking of orthogonal vectors and Maxwell's equations and vector products.
I was wondering, in what sense, physically speaking, are the electric and magnetic fields perpendicular to each other?
I have seen that electromagnetism is the result of relativistic effects at the quantum level. Is it that the rotation of an electron creates these effects, and magnetic effects are related to the polar axis spin of the electron, and electric effects to the equatorial rotation of the electron? And this is why the two fields are perpendicular to each other?
And why from one frame of reference, you can see an electric field, and from another frame of reference you see a magnetic field - because it depends on your angle relative to the electron's motion?
 A: 
And why from one frame of reference, you can see an electric field, and from another frame of reference you see a magnetic field - because it depends on your angle relative to the electron's motion?

No, it does not depend on the angle of motion, but on the inertial frames.
Special relativity is necessary to give the mathematical formulation.


Lorentz boost of an electric charge.


Top: The charge is at rest in frame F, so this observer sees a static electric field. An observer in another frame F′ moves with velocity v relative to F, and sees the charge move with velocity −v with an altered electric field E due to length contraction and a magnetic field B due to the motion of the charge.


Bottom: Similar setup, with the charge at rest in frame F′.

A: The electric and magnetic fields are generally not perpendicular.  Presumably you're thinking of electromagnetic waves propagating in vacuum, in which case the electric and magnetic fields are perpendicular to each other and to the direction of propagation of the wave.  But of course, this is not always true; in particular, it's easy to create electric and magnetic fields in the laboratory which point in whatever directions you'd like.
Beyond this, it seems like you're overthinking the issue.  The electric and magnetic fields have directions associated to them, and in certain cases those directions are perpendicular to one another.  There is no reason whatsoever to invoke quantum mechanics, electron spin, or relativity when talking about that fact.
A: If there is an inertial frame where $\bf{E}=0$ (or $\bf{B}=0$), than in all other inertial frames will be either $\bf{E}=0$ (or $\bf{B}=0$), or $\bf{E} \perp \bf{B}$.
If, for instance, $\bf{B}=0$, than in this frame the charge is accelerated in the direction of $\bf{E}$. But from the other frame this dynamics is seen as combination of acceleration and rotation. This is because in Minkowski spase all that happens to vectors is either acceleration (change of energy and magnitude of spatial momentum) or rotation (change in the direction of spatial momentum).
Any acceleration (Lorentz boost) is interpreted as due to (transformed) $\bf{E}$, while the (3D) rotation is interpreted as due to (transformed) $\bf{B}$. That's why in the new frame we can see the field ($\bf{B}$ or $\bf{E}$) that was zero in the initial frame. But they will always be orthogonal due to transformation properties of the acceleration under Lorentz transformation, regardless of the properties of the sources of the fields $\bf{B}$ and $\bf{E}$.
On the pther hand, if $\bf{E} \perp \bf{B}$ and $E \neq B$, than there is an inertial frame where $\bf{E}=0$ or $\bf{B}=0$. The only case when fields are orthogonal in all frames is  $\bf{E} \perp \bf{B}$ and $E = B$ (elecrtomagnetis waves in vacuum).
If the fields are not orthogonal in one frame, they will not be orthogonal in any other frame. In that case there is a reference frame where both $\bf{B}$ and $\bf{E}$ are parallel to each other, and acceleration (due to change in energy and magnitude of momentum) and centripetal (or centrifugal) force have the same direction in this frame.
A: 
found mathematical explanations speaking of orthogonal vectors and Maxwell's equations and vector products

This is true and these mathematical explanations are also consistent with what we find physically. If you accept that Maxwell's equations consistently describe electromagnetism and lead to electromagnetic waves (and they do), you can show that $$\bf E\cdot B=0$$ for an electromagnetic wave. That is, the $\bf B$ and $\bf E$ fields are orthogonal not only as a mathematical consequence, but this also corresponds to how they behave physically (for electromagnetic waves in a vacuum).

in what sense, physically speaking, are the electric and magnetic fields perpendicular to each other?

In the sense that the $\bf E$ and $\bf B$ fields, physically  oscillate at $90^\circ$ to each other and at $90^\circ$ to the direction of propagation. That is physically how it is (again, for electromagnetic waves in space).

I have seen that electromagnetism is the result of relativistic effects at the quantum level. Is it that the rotation of an electron creates these effects, and magnetic effects are related

You do not need to go into relativity or quantum mechanics. You are adding a layer of complexity that is not needed for describing this aspect of electromagnetic waves.
Edit: As pointed out in the comments below, in the general case of electric and magnetic fields, these fields need not be orthogonal.
A: I believe the best way to visualize physically your question is in the case of the electromagnetic field of a current carrying wire conductor.
The explanation given must be fundamental and must include the dressed electron field of the bare electron's mass which is the origin and source of electromagnetism phenomenon. The discrete free electron drifting inside the wire, its electromagnetic flux envelope is cascaded with that of the other electrons inside and along the wire as  illustrated below in fig.1:

Important here is to understand shown in fig.1 that the single electron does not have separate electric and magnetic flux but a unified electromagnetic flux manifold. The electromagnetic quantum flux (i.e. EM flux of the electron quanta) of all the cascaded coherently electrons inside the current carrying wire generate the uniform macroscopic electric field inside along the wire E and magnetic B field envelope outside and along the wire. Both constitute the electromagnetic macroscopic field of the current carrying wire.
You can clearly see in the above illustration (on the right illustration of fig.1, conventional flow of current is used), that axially the electric field vector E inside the wire which is in the same direction of current I Poynting vector in the case of a current carrying wire (conventional flow of current is used), is perpendicular to the magnetic B field vectors outside the wire.
This perpendicular characteristic of the B and E components of the electromagnetic field travelling along a wire generating electric current is an inherent property of the dressed electron electromagnetic flux envelope.
On the left side of the fig.1 illustration we see the cascaded electron's field manifolds central electromagnetic flux (see horn tube flux formation diametrically on each electron manifold) clearly forming the E field inside the wire and the outside periphery electromagnetic flux of each electron manifold forming the B uniform macroscopic field outside the wire.
Notice also importantly, how the interchangeable between the electric and magnetic components nature of the electromagnetism phenomenon is demonstrated in fig.1?
The cascaded quantum flux of each electron manifold vertical segment (see diametrical horn tube formation of each manifold) which represents the magnetic moment of each electron becomes the macroscopic net electric field E inside the wire conductor and the periphery quantum flux of each electron manifold which represents the electron charge, becomes the macroscopic magnetic field B outside the wire.
The bare electron mass is positioned at the center of the dressed electron field manifold shown in fig.1, at a dimensionless point. Therefore the bare electron model is characterized by the known literature as an elementary dimensionless-point massive particle which has created in the past much confusion and non-intuitive understanding. The actual physical form of the electron particle is dressed meaning it has a charge radius.
This concludes IMO the physical explanation you particular asked in your question, besides the other more formal correct answers given.
