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Hyperphysics states that, unlike an electron, the helicity of a neutrino is invariant because we cannot change to a reference frame where it is different:

This and subsequent experiments have consistently shown that a neutrino always has its intrinsic angular momentum (spin) pointed in the direction opposite its velocity. It is called a left-handed particle as a result. Anti-neutrinos have their spins parallel to their velocity and are therefore right-handed particles. Therefore we say that the neutrino has an intrinsic parity. [...] For neutrinos, however, which are traveling at the speed of light or very close to it, you cannot accelerate to a greater speed and thereby change their "handedness".

Is this fundamentally true though? My understanding is that neutrinos have small mass and therefore travel slightly slower than light. Or is it just practically true, because we can't (yet) move faster than them into a reference frame where the helicity is reversed?

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It sounds like that site is discussing the Standard Model neutrino. Neutrinos were presumed to be massless for a long time and the SM still models them as massless.

We now know that this is not true, that (at least two of the three flavors of) neutrinos do have a small mass, although the SM is still a good approximation. So, like other massive particles it is indeed possible in principle to look at them in a reference frame where their helicity is reversed.

This has not yet been incorporated into the SM, in part because we don't know whether neutrinos are Dirac particles or Majorana particles. Here's a short article that talks about it a bit more.

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