  We point out a novel possibility for neutrinos where all neutrino flavors can
be part Dirac and part Majorana. Our primary motivation for this model comes
from an attempt to use supersymmetric see-saw models to tie inflation, baryon
asymmetry of the Universe and dark matter to the neutrino sector. The idea
however could stand on its own, with or without supersymmetry. We present a
realization of this possibility within an $S_3$ family symmetry for neutrino
masses, where we obtain tri-bi-maximal mixing for neutrinos to the leading
order. The model predicts that for the case of inverted hierarchy, the lower
limit on the neutrino mass measured in neutrinoless double beta decay
experiments is about a factor of two larger than the usual case.
