  Since the $\nu_\tau(\bar\nu_\tau) A_Z \to \tau^\mp X$ reaction is notoriously
difficult to be directly measured, the information on the dynamics of this
nuclear process should be extracted from the analysis of the energy and angular
distributions of the tau decay visible products. These distributions depend on
the components of the tau-polarization vector. We give, for the first time, the
general expression for the outgoing hadron (pion or rho meson) energy and
angular differential cross section for the sequential $\nu_\tau A_Z \to
\tau^-(\pi^- \nu_\tau, \rho^-\nu_\tau) X$ and $\bar\nu_\tau A_Z \to
\tau^+(\pi^+ \bar\nu_\tau, \rho^+ \bar\nu_\tau) X$ reactions. Though all
possible nuclear reaction mechanisms contribute to the distribution, it may be
possible to isolate/enhance one of them by implementing appropriate selection
criteria. For the case of the quasi-elastic reaction off oxygen and neutrino
energies below 6 GeV, we show that the pion distributions are sensitive to the
details of the tau-polarization components. We find significant differences
between the full calculation, where the longitudinal and transverse components
of the tau polarization vector vary with the energy and the scattering angle of
the produced tau, and the simplified scheme in which the polarizations are set
to one and zero (respective asymptotic values in the high energy regime). In
addition to its potential impact on neutrino oscillation analyses, this result
can be used to further test different nuclear models, since these observables
provide complementary information to that obtained by means of the inclusive
nuclear weak charged-current differential cross section. We also study the
effects on the cross section of the $W_4$, $W_5$ nuclear structure functions,
which contributions are proportional to the charged lepton mass, and therefore
difficult to constrain in muon and electron neutrino experiments.
