[Eric A. Taylor (OP)]

Are there particles that are counterparts to the neutrino, like protons and electrons to the neutrons?

[Soul Surfer]

you are not thinking correctly here.   The electron and positron are linked with the neutrino as leptons or light particles and are not considered to be composite.

The proton anti proton and neutron (with its antiparticle) are Baryons or heavy particles made up of quarks