  The semitauonic $B_c^- \to J/\psi \tau^-\bar{\nu}_\tau$ decay is optimal to
scrutinize possible new physics effects in $b \to c \tau^- \bar{\nu}_\tau$
transitions as indicated by the current data on $R(D^{(*)})$ anomalies. In this
work, we study the $B_c^- \to J/\psi \tau^-\bar{\nu}_\tau$ decay with both
polarized $\tau$ and $J/\psi$. Their subsequent decays, with $J/\psi\to \mu^+
\mu^-$ as well as $\tau^- \to \pi^- \nu_\tau$, $\tau^- \to \rho^- \nu_\tau$ and
$\tau^- \to \ell^-\bar{\nu}_\ell\nu_\tau$, are exploited to extract the energy
and angular distributions of the charged final-state particles in the
processes. Starting with the most general effective Hamiltonian relevant for
the $b \to c \tau^- \bar{\nu}_\tau$ transitions, including all possible Lorentz
structures of the dimension-six operators with both left- and right-handed
neutrinos, we first derive the five-fold differential decay rate in terms of
the visible final-state kinematics. From this distribution, we then construct
in total 34 normalized observables, among which nine refer to the CP-violating
triple product asymmetries that vanish within the Standard Model. We also
construct five new observables based on the combinations of these normalized
observables that can only be attributed to the right-handed neutrinos. On the
other hand, considering the low statistics of the fully differential
distribution, we introduce some integrated observables with only one kinematic
variable left, which are more promising to be measured due to the largely
increased statistics. The sensitivities of all these observables to the
different new physics scenarios are investigated in detail. Finally, assuming
an ideal circumstance, we give an estimate of the statistical uncertainties of
the nine CP-conserving observables at LHCb and found that $\tau^-\to
\pi^-\nu_\tau$ has the highest analyzing power among the three $\tau$ decay
channels.
