Reading this question and its answer
The diagonalization of the see-saw is described as going from "interaction basis" right (and left) handed neutrinos to "mass basis" right (and left) handed neutrinos.
Now, can chirality refer to a particular basis? Is it not a general property connected with parity? I would think that our "left handed neutrino" after seesaw is not a left neutrino anymore, but that it has a small mixing with the "right handed neutrino".
EDIT: and if it is so... is it always so, or there is some see-saw mechanism that preserves chirality?