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.