  We study the neutrino spin oscillations, i.e., neutrino spin precession
caused by the neutrino interaction with matter polarized by external magnetic
field (or, equivalently, by the interaction of the induced magnetic moment
(IMM) of a neutrino with the magnetic field). In the analysis, we consider
realistic conditions inside supernovae and discuss both the Dirac and Majorana
cases. We show that due to the interaction with the polarized matter a neutrino
flux from a supernova suffers additional attenuation at low neutrino energies.
Another possible effect is that when taken together the effects of conventional
magnetic moment and polarized matter can cancel each other so that under
certain condition the oscillations disappear. We note that this can lead to the
appearance of a characteristic maximum in the spectrum of electron neutrinos
from supernovae. Accounting for neutrino mixing (in the two-flavor
approximation) can significantly increase (almost twice) the effective value of
the IMM of the electron neutrino. As a result, electron neutrinos with twice as
high energy will be able to participate in the processes of spin conversion.
Various feedback mechanisms accompanying the considered phenomena are also
discussed.
