What mechanism creates the difference between left and right in the universe? Or, more general, what mechanism creates the distinction between two opposite directions in the universe, which manifests itself in the existence of only left-handed neutrinos?
The standard model offers no explanation but gives us only a description. 
 A: 
What mechanism creates the difference between left and right in the universe?

None.  

Or, more general, what mechanism creates the distinction between two opposite directions in the universe.

None. 

which manifests itself in the existence of only left-handed neutrinos? 

None. Because a right-handed neutrino is an anti-neutrino. See hyperphysics:

Re the comments, note that parity is nothing special. Don't expect to be able to reflect the Wu experiment or a current loop in a mirror.
 
Image1 by nagualdesign, see Wikipedia, image2 courtesy of Sbyrnes321, see Wikipedia

The standard model offers no explanation but gives us only a description. 

Because it's incomplete. IMHO it's much more incomplete than people think. It doesn't tell you what a photon is, or what an electron is, or what a proton is, or what a neutron is, or what a neutrino is. But I can tell you this: the electron is not a point particle. Nor are any other particles. Hence the non-local hidden variables. No particle is local. A particle can be localized by an interaction, but do not forget that the λ in E=hc/λ stands for wavelength. A photon has a wavelength. We can diffract photons. And electrons, and neutrons. 
PS:IMHO you shouldn't get distracted by the Rishon model, there's no evidence for it.    
