  In this paper a scenario with the participation of the exotic scalar S,
tensor T and pseudoscalar couplings of the right-handed neutrinos in addition
to the standard vector V, axial A couplings of the left-handed neutrinos in the
low-energy $(\nu_{\mu}e^{-})$ and $(\nu_{e}e^{-})$ scattering processes is
considered. Neutrinos are assumed to be massive Dirac fermions and to be
polarized. Both reactions are studied at the level of the four-fermion point
interaction. The main goal is to show that the physical consequence of the
presence of the right-handed neutrinos is an appearance of the azimuthal
asymmetry in the angular distribution of the recoil electrons caused by the
non-vanishing interference terms between the standard and exotic couplings,
proportional to the transverse neutrino polarization vector. The upper limits
on the expected effect of this asymmetry for the low-energy neutrinos $(E_{\nu}
< 1 MeV)$ are found. We also show that if the neutrino helicity rotation
$(\nu_{eL} \to \nu_{eR})$ in the solar magnetic field takes place, the similar
effect of the azimuthal asymmetry of the recoil electrons scattered by the
solar neutrinos should be observed. This effect would also come from the
interference terms between the standard $(V, A)_{L}$ and exotic $(S, T, P)_{R}$
couplings. New-type neutrino detectors with good angular resolution could
search for the azimuthal asymmetry in event number.
