  For a general $H_b\to H_c\tau\bar\nu_\tau$ decay we analyze the role of the
$\tau$ polarization vector ${\cal P}^\mu$ in the context of lepton flavor
universality violation studies. We use a general phenomenological approach that
includes, in addition to the Standard Model (SM) contribution, vector, axial,
scalar, pseudoscalar and tensor new physics (NP) terms which strength is
governed by, complex in general, Wilson coefficients. We show that both in the
laboratory frame, where the initial hadron is at rest, and in the center of
mass of the two final leptons, a $\vec {\cal P}$ component perpendicular to the
plane defined by the three-momenta of the final hadron and the $\tau$ lepton is
only possible for complex Wilson coefficients, being a clear signal for physics
beyond the SM as well as time reversal (or CP-symmetry) violation. We make
specific evaluations of the different polarization vector components for the
$\Lambda_b\to\Lambda_c$, $\bar B_c\to\eta_c,J/\psi$ and $\bar B\to D^{(*)}$
semileptonic decays, and describe NP effects in the complete two-dimensional
space associated with the independent kinematic variables on which the
polarization vector depends. In this way, we go beyond previous analyses, where
only some angular averages, which appear in the study of the subsequent
hadronic decay of the produced $\tau-$lepton, have been considered. We find
that the detailed study of ${\cal P}^\mu$ has great potential to discriminate
between different NP scenarios for $0^-\to 0^-$ decays, but also for $\Lambda_b
\to \Lambda_c$ transitions. For this latter reaction, we pay special attention
to corrections to the SM predictions derived from tensor (real and complex)
contributions.
