[not2b]

I've seen another argument, but I lack the competence to assess its validity: if a (left-handed) neutrino has mass, it moves at less than the speed of light, which means you can pass it and look back at it. You'd then see it as having reversed spin. But that might be based on a classical physics analogy that doesn't hold.

[SaberTail]

That's correct. The more jargon-y way to say it would be that for massive particles, helicity isn't Lorentz invariant.

It's an approximation that helicity equals chirality (which is what matters for weak force interactions), but this approximation is pretty good for particles moving close to the speed of light (which neutrinos tend to do, due to their low mass)

[T-A]

You're thinking about helicity, not chirality:

https://en.wikipedia.org/wiki/Neutrino#Chirality