In fact, in the presence of matter (and, importantly, anisotropies in the background matter field), neutrinos can change Chirality. This fact can be derived from careful analysis of the Quantum Kinetic Equations, see:  and 
Due to the nature of this Chiral flipping, the amplitude to get such a flip is suppressed from the usual weak-couplings (of neutrinos on matter) by another order of $m_\nu/E_\nu$ and so it seems difficult to get a significant effect.
There has been some explorations of whether this Chiral flipping occurs in astrophysical environments, but it seems that one needs either an unrealistically large neutrino rest mass to get this to happen, and/or an extremely flat electron-fraction profile that hovers near $Y_e\approx 1/3$ (the resonance condition for such an effect). See for example:  ,  ,  , 
In addition to this kinetic effect (caused by scatterings), one can also obtain Chiral flipping due to the (extremely small, but present) magnetic moment of neutrinos. Large magnetic fields can perhaps also give rise to this effect. See, for example:  ,