Do point out any mistakes. I don't pretend to know this properly!
Neutrinos are weird. I don't really know much about them. They
oscillate between different states as they go and presumably antineutrinos oscillate between the three corresponding antiparticle states. To make things weirder neutrinos and antineutrinos may actually be the same thing. Or may not. I don't know whether the mass discrepancy in the link budgenator provided would mean neutrinos and antineutrinos were definitely not Majorana particles.
Neutrinos aren't rare, they just interact very, very weakly with everything else, on account of being uncharged (neutr-) and having tiny mass (-ino). A second after it is produced by radioactive decay a neutrino is likely to be well outside lunar orbit having largely failed to notice the Earth. They're not really the annihilating types.
I keep meaning to learn this stuff. By the time we reached the complicated particle stuff at university I was seriously depressed and had trouble staying awake in lectures, not that I'd've understood it anyway...
...I vaguely recall drawing the funny lopsided house-of-cards diagrams with lots of particles on...
Sounds very interesting. I myself do not know much about the behavior of particles. I'll have to do more research about this. Perhaps ruveyn could shed a bit more light as he appeared to me as very knowledgeable in other threads I've seen on such topics.