  We consider the most general neutrino masses and mixings including Dirac mass
terms, M_D, as well as Majorana masses, M_R and M_L. Neither the Majorana nor
the Dirac mass matrices are expected to be diagonal in the eigenbasis of weak
interactions, and so the resulting eigenstates of the Hamiltonian are
admixtures of $\SU(2)_L$ singlet and doublet fields of different
``generations.'' We show that for three generations each of doublet and singlet
neutrinos, diagonalization of the Hamiltonian to obtain the propagating
eigenstates in the general case requires diagonalization of a $12\times12$
Hermitian matrix, rather than the traditional $6\times6$ symmetric mass matrix.
The symmetries of the $12\times12$ matrix {\em are} such that it has 6 pairs of
real eigenvalues. Although the standard "see-saw" mechanism remains valid, and
indeed the eigenvalues obtained are identical to the standard ones, the correct
description of diagonalization and mixing is more complicated. The analogs of
the CKM matrix for the light and the heavy neutrinos are nonunitary, enriching
the opportunities for CP violation in the full neutrino sector.
