Neutrino Helicity and Spin Uncertainty

[Coltrane8]

TL;DR Summary

Can I change helicity by measuring Spin in the perpendicular direction of motion, in accordance to commutation rules and uncertainty of Spin operators?

In the context of non relativistic quantum mechanics, or better, if I consider the neutrino's mass to be zero, the phrase

Neutrino are left-handed. Choosing $\hat{z}$ axis as the direction of motion, the spin projection onto that axis is $- \hbar/2$. In this sense the direction of motion is opposite to spin.

seems to me puzzling. What I know is that if I know the direction of motion, I know the spin projection onto that direction, say $\hat{z}$-direction. But to not violate Heisenberg's Uncertainty, if I manage to measure Spin in a perpendicular direction of motion, say $S_x$, now helicity must be undetermined. And to be consistent if I subsequently measure $S_z$ (or helicity), I can have the same or the opposite helicity. So, if a measure change helicity (to not to violates Heisenberg), it would be more precise to talk about left-handed neutrinos only in the context of some interaction, without excluding a priori that I can change it's helicity by an accurate measurement of Spin on the perpendicular direction of motion?

 

[PeroK]

You may get a more expert answer, but I don't think you can apply the $S_x$ operator to a relativistic particle and hope to get eigenstates of the Dirac spinor.