Conceptual interpretation of the left- and right-handed spinor representations of the Lorentz group I understand mathematically that the Lorentz group's Lie algrebra $\mathfrak{so(3,1)}$ (given by eqns. (33.11)-(33.13) in Srednicki's QFT book) is isomorphic to $\mathfrak{su(2) \times su(2)}$ (given by eqns. (33.18) - (33.20)), and thus that the Lorentz group has two inequivalent irreducible representations with $n + n' = 1/2$.  But I don't understand the distinction between these representations at the conceptual level - Srednicki's describes the $N$ operators only as "nonhermitian operators whose physical significance is obscure."
How should I picture the difference between a left-handed and right-handed spinor field?  The question Explaining chirality for spin 1/2 particle does a nice job distinguishing helicity from chirality, but ideally I'd like an explanation of chirality that makes no reference to helicity at all.  (In my experience, most explanations of chirality start by pretending it's the same thing as helicity, then go on to clarify that they're actually different for massive particles, but never actually go on to properly define chirality except in a formal mathematical way.  I feel that it's conceptually clearer to just explain what chirality is rather than explaining what it isn't.)
For example, conceptually why does the parity operator reverse a particle's chirality?  (I know that this is usually just treated as part of the parity operator's definition.)  If one were to be given a particle without being told its chirality, how would one check it?  For example, if I were to consider a single Weyl field with a Lagrangian given by Srednicki's eqn. (36.2),
$$ \mathcal{L} = i \psi^\dagger \bar{\sigma}^\mu \partial_\mu \psi - \frac{1}{2} m \left( \psi \psi + \psi^\dagger \psi^\dagger \right),$$
how could I determine its chirality experimentally?
 A: I'm answering this question very late because I've noticed that a lot of chirality/helicity questions around this site have rather poor answers (though the other answer on this question is excellent!), so this is to help clear the air.
As with most issues surrounding chirality and helicity, the problem is dissolved as long as one remembers that chirality is a property of fields and helicity is a property of particles, and fields are tools to create and annihilate particles. 
For example, as stated in great detail here, a right-chiral Weyl field is one which annihilates certain left-helicity massless particles and creates certain right-helicity massless particles, which have opposite charges. The same goes for left-chiral Weyl fields with left and right swapped. 

How should I picture the difference between a left-handed and right-handed spinor field?  

In terms of classical fields, a right-handed field is one which has plane wave solutions where the eigenvalues under rotation about the axis and translation along the axis have the same sign, while a left-handed field is the opposite. 
But this picture isn't too useful in quantum field theory. At this level, fields are a tool for bookkeeping the creation and annihilation of particles. A left-handed and right-handed Weyl spinor field correspond to precisely the same particle content, so there isn't a physical difference. For instance, a left-handed Weyl field of positive charge is associated with the same particles as a right-handed Weyl field is negative charge; one creates what the other annihilates. 

The question Explaining chirality for spin 1/2 particle does a nice job distinguishing helicity from chirality, but ideally I'd like an explanation of chirality that makes no reference to helicity at all.

Fields transform in representations of the Lorentz group. If one extends this to the full Lorentz group, which includes parity, representations are chiral if they don't map to themselves under parity. 

For example, conceptually why does the parity operator reverse a particle's chirality? 

It doesn't, because chirality is a property of fields, not particles. At the level of the fields, parity flips chirality essentially by definition, if you use the definition I gave above.

If one were to be given a particle without being told its chirality, how would one check it?  For example, if I were to consider a single Weyl field with a Lagrangian given by Srednicki's eqn. (36.2),
  $$ \mathcal{L} = i \psi^\dagger \bar{\sigma}^\mu \partial_\mu \psi - \frac{1}{2} m \left( \psi \psi + \psi^\dagger \psi^\dagger \right),$$
  how could I determine its chirality experimentally?

Again, this is a meaningless question because chirality is a property of fields, which are used to organize a theory of particles. For example, suppose you detect certain right-helicity particles, and left-helicity particles with the opposite charges, all of which are massless. You can describe them using a Lagrangian containing a left-chiral Weyl field, or a Lagrangian containing a right-chiral Weyl field, or even a Lagrangian containing a Dirac field obeying a constraint. 
It's kind of like saying, if you detect a photon, can you tell if the associated field is $A_\mu$ or $F_{\mu\nu}$ or something else. The question isn't really meaningful. Either one or both of these fields might appear in the process of describing the very same physical observations of the particle.
