  In this paper, we indicate a possibility of utilizing the elastic scattering
of the Dirac low energy ($\sim 1$ MeV) electron neutrinos ($\nu_e$'s) on the
polarized electron target (PET) in testing the time reversal symmetry violation
(TRSV). We consider a scenario in which the incoming $\nu_e$ beam is the
superposition of left chiral (LC) and right chiral (RC) states. LC $\nu_e$'s
interact mainly by the standard $V - A$, while RC ones are only detected by the
exotic $V + A$ and $S_R, P_R, T_R$ interactions. One assumes that the spin
polarization vector of the initial $\nu_e$'s is turned aside from the $\nu_e$
momentum, and due to this the non-vanishing transversal component of the
$\nu_e$ spin polarization appears. We compute the differential cross section as
a function of the recoil electron azimuthal angle and scattered electron
energy, and show how the interference terms between standard $V-A$ and exotic
$S_R, P_R, T_R$ couplings depend on the various angular correlations among the
transversal $\nu_e$ spin polarization, the polarization of the electron target,
the incoming neutrino momentum and the outgoing electron momentum in the limit
of relativistic $\nu_e$. We illustrate how the maximal value of recoil
electrons azimuthal asymmetry and the asymmetry axis location of outgoing
electrons depend on the azimuthal angle of the transversal component of the
$\nu_e$ spin polarization, both for the time reversal symmetry conservation
(TRSC) and TRSV. Next, we display that the electron energy spectrum and polar
angle distribution of the recoil electrons are also sensitive to the
interference terms between $V - A$ and $S_R, P_R, T_R$ couplings, proportional
to the T-even and T-odd angular correlations. Our model-independent analysis is
carried out for the flavor $\nu_e$'s.
