A thought experiment about neutrinos I don't understand all the details of Dirac mass, Majorana mass, and many other "deep" notions.
I have in mind a very simple thought experiment.
Because of neutrino oscillations we know neutrinos have mass. Thus their speed is less than $c$.
I imagine a beam of neutrinos created by some experiment in a lab. They are neutrinos, not antineutrinos, and have energy much larger than their rest mass. So they have left-handed helicity.
Now I imagine (this is a thought experiment, OK?) that some lab is moving, with respect to the one that created them, at a speed so very, very close to $c$ that it will overtake the beam, so fast that in the frame of this second lab, the speed of the beam appears to be directed towards the lab at a speed close to $c$ and in fact, opposite of their speed in the frame of the lab that created them.
In the frame of this new lab, the particles that are directed towards it have right-handed helicity.
Now two things can happen:
A) either they interact with the instruments in that lab with the same efficiency as in the original lab, and this means, since they have right-handed helicity, that they are now antineutrinos, as seen in this lab. So lepton number is not conserved.
B) or lepton number is conserved, they are still neutrinos, but having the right-handed helicity, which means the "wrong one" for neutrinos, they would interact much, much less than neutrinos of the correct, left-handed helicity. Then they are, in that lab, sterile neutrinos, but their rest mass is the same as for "normal neutrinos" and this sounds wrong.
So which is which? And please, don't throw me complicated notions that I cannot follow, Dirac vs Majorana mass, symmetry groups, chiral anomalies, etc.
Just tell me, A is right or B is right.
Thanks.

Well, thanks to you folks, I have learned something. I really mixed up chirality and helicity, and that has been cleared up.
I upvoted all of you, but I cannot accept an answer to a question so ill-posed.
But your answers only bring more questions.
Rather than editing this question, I think it would be better to ask a new one. I have to digest all this before asking a well-posed question (I hope).
If, by the time  I am ready, from your by answers or comments, I see a consensus that I should edit it rather than ask a new one, I shall oblige.

OK, so here is where my new question is.
 A: Handedness is not about helicity - which, as the question points out, is something that can change depending on the reference frame - but about chirality (see also this answer of mine). The two notions are only the same for massless particles - for which then helicity is also frame-invariant since massless particle have no rest frame.
So when you try to outrun a neutrino, nothing interesting happens. Its helicity flips, sure, but its chirality - the thing the weak interaction cares about - doesn't.
If you know a bit more about massive particles and their chirality, you might now worry whether or not the propagating neutrino then is really purely left-handed - usually massive fermions propagate as a mixture of chiralities. However, this is only true for fermions with a "Dirac mass", while those with a "Majorana mass" don't show this behaviour -  see this question and its answers for a discussion of whether the neutrino can have a Majorana mass. And since we can't see the right-handed part anyway if it is truly sterile, it's hard to get any experimental clarification on this except if we actually do find a right-handed neutrino somewhere.
A: You're confusing helicity with chirality. Chirality is Lorentz invariant, i.e. it cannot be changed by overtaking. When people talk about left-handed neutrinos, they mean that they are left-chiral. Right-chiral neutrinos do not exist (or as hypothesized sterile neutrinos, so called because they do not interact weakly). It hence is chirality that dictates behaviour in interactions.
There is nothing wrong with neutrinos having right helicity.
A: This is the question about neutrino masses.
If you outrun a left-handed neutrino, and the right-handed particle interacts like an antineutrino, then the neutrino is its own antiparticle.
If you outrun a left-handed neutrino, and the right-handed particle is sterile because the weak interaction doesn’t couple to right-handed matter particles and the matter neutrino doesn't couple (at tree level) to electromagnetism or the strong force, then the neutrino and the antineutrino are different.
The problem is that both of these possibilities are consistent with all of the neutrino data that we have so far. Nobody knows the answer to your question.

Since there are now several comments and answers pointing out that chirality, unlike helicity, is invariant under boosts, I should clarify. A particle with Dirac-type mass cannot have definite chirality in its rest frame, and therefore cannot have constant chirality in any other frame. I was implicitly assuming that "outrunning" the neutrino took some finite amount of time from its creation in a left-chiral state, and that the neutrino in its rest frame had evolved into an incoherent mixture of left- and right-chiral components.
You can absolutely change whether (or better, how much) a particle with Dirac mass participates in the weak interaction by flipping its helicity; I spent fifteen years doing this with polarized beams of electrons and neutrons.
