  The helicity dependence of the inclusive $\pi^-$ photoproduction reaction
from the deuteron in the $\Delta$(1232)-resonance region is investigated with
inclusion of final-state nucleon-nucleon rescattering ($NN$-FSI). For the
elementary $\pi$-production operator an effective Lagrangian model which
includes the standard pseudovector Born terms and a contribution from the
$\Delta$-resonance is used. The half-off-shell $NN$-scattering matrix is
obtained from a separable representation of a realistic $NN$-interaction. The
differential polarized cross-section difference for parallel and antiparallel
helicity states is predicted and compared with experiment. We find that the
effect of $NN$-FSI is much less important in the helicity difference than in
the previously studied unpolarized differential cross section. Furthermore, the
contribution of $\vec d(\vec\gamma,\pi^-)pp$ to the deuteron spin asymmetry is
explicitly evaluated with inclusion of $NN$-FSI. It has been found that the
effect of $NN$-FSI is much larger in the asymmetry than in the total cross
section, and this leads to an appreciable reduction of the spin asymmetry in
the $\Delta$-region. Inclusion of such effect also leads to improved and quite
satisfactory agreement with existing experimental data.
