Why does the weak force distinguish left and right handedness? I'm wondering why the weak interaction only affects left-handed particles (and right-handed antiparticles).
Before someone says "because thats just the way nature is" :-), let me explain what I find needs an explanation:
In the limit of massless fermions, chirality (handedness) becomes helicity $(\vec S \cdot \hat p)$. Now, helicity is a property of the state of motion of an object in space. It is pretty unobvious to me how the internal symmetry $SU(2) \times U(1)$ would "know" about it, and be able to distinguish the two different helicity states of motion.
On a more technical level, IIRC, left and right handed spinors are distinguished by their transformation properties under certain space-time transformations, and are defined independent of any internal symmetry. If we want to get the observed V-A / parity violating behavior, we have to plug in a factor of $(1 - \gamma^5)$ explicitly into the Lagrangian.
Is there any reason this has to be like this? Why is there no force coupling only to right handed particles? Why is there no $(1 + \gamma^5)$ term? Maybe it exists at a more fundamental level, but this symmetry is broken?
 A: The explanation is simple--- all particles we can see are chiral, they have only one handedness, because if they had both handedness, they could get a mass, and generically, that mass would be of order of magnitude the Planck mass. We live at energy scales which are teeny compared to the Planck mass, so we can only see massless stuff, so we only see chiral fermions (and gauge bosons).
The right question then is the other way around, if everything is chiral, why then do the electromagnetic and strong interactions not violate parity?
This is because the Higgs mechanism partners up the chiralities into massive Dirac particles at lower energies, and only the W,Z bosons know that they were chiral to begin with. At low energies, you get parity and charge-conjugation as accidental symmetries, because these are symmetries of the low-energy Dirac Lagrangian coupled to the remaining photon and gluons.
As for the neutrinos, a chiral neutrino can have a Majorana mass while only having one chirality, and this is certainly what is going on in nature, since this scheme predicts the mass correctly, and this mass is absurdly small.
A: There's nothing a priori that says it has to be that way and (I believe) neutrino oscillations have now shown that they have a mass which implies both right and left handed neutrinos exist.  That said, the lagrangian is still seriously skewed towards the left handed interaction.  So it's still very left-handed in a sense but there's nothing that says a coupling to right handed particles is out.
We just don't see a strong coupling experimentally.  
A: I think you are sort of reversing the logic of chirality and helicity in the massless limit. Chirality defines which representation of the lorentz group your Weyl spinors transform in.  It doesn't 'become' helicity, helicity 'becomes' chirality in the massless limit. That is, chirality is what it is, and it defines a representation of a group and that can't change. This other thing we have defined called helicity just happens to be the same thing in a particular limit.
Now once you take the massless limit the Weyl fermions are traveling at the speed of light you can no longer boost to a frame that switches the helicity. I think its best to think of a fermion mass term as an interaction in this case and remember that the massive term of a Dirac fermion is a bunch of left and right- handed Weyl guys bumping up into one another along the way. Conversely if you want to talk about a full massive Dirac fermion that travels less than c and you can boost to change the helicity, but that full Dirac fermion isn't the thing carrying weak charge, only a 'piece' of it is. 
See this blog post on helicity and chirality. 
As far as the left-right symmetry being broken people have certainly built models along these lines but I don't think they have worked out.
Does this answer your question?
A: This answer is similar to Ron Maimon's but maybe it will be helpful.
In short, the weak force doesn't violate parity; only the fermions do.
There are no Dirac fermions in the Standard Model. There are no chiral projections in the Lagrangian because there's nothing for them to act on. There are no gamma matrices at all, just two-component Weyl spinors and Pauli matrices.
At low temperature, there are some combinations of Weyl fields that behave for many practical purposes like Dirac fermions. If you see a Standard Model Lagrangian with Dirac spinors in it, it's one in which the Weyl fields have been paired up "in anticipation of" this low-energy approximate description being useful. The chiral projection operators are there to split them apart again where necessary. Technically, they should never have been joined in the first place. The weak force isn't doing the projecting; it's a human invention.
The Dirac-like behavior comes from Yukawa couplings involving the Higgs field. The Higgs field has SU(2) and U(1) charge but no SU(3) charge, and as a result the only possible gauge-invariant Yukawa couplings are between fermion fields whose SU(3) charges match and whose SU(2) and U(1) charges don't. The gauge forces don't "know" about any of this; they couple to whatever has charge, but because of the way the quasi-Dirac fields are coupled together, their halves have mismatched SU(2) and U(1) charges, but equal SU(3) charges.
As mentioned in Ron Maimon's answer, a plausible reason why it "has to be like this" is that there's nothing to prevent a direct coupling between mirror-symmetric fermion fields, i.e., a real mass term, and no known reason why the resulting mass wouldn't be comparable to the Planck mass. So we probably wouldn't see symmetric fermion fields even if they exist. Why asymmetric fermions do exist is an open question. I think that many otherwise promising approaches to quantum gravity have had to be abandoned because they can't accommodate the chiral fermions.
