  The first full angular analysis of the $B^0 \to D^{*-} D_s^{*+}$ decay is
performed using 6 fb$^{-1}$ of $pp$ collision data collected with the LHCb
experiment at a centre-of-mass energy of 13 TeV. The $D_s^{*+} \to D_s^+
\gamma$ and $D^{*-} \to \bar{D}^0 \pi^-$ vector meson decays are used with the
subsequent $D_s^+ \to K^+ K^- \pi^+$ and $\bar{D}^0 \to K^+ \pi^-$ decays. All
helicity amplitudes and phases are measured, and the longitudinal polarisation
fraction is determined to be $f_{\rm L} = 0.578 \pm 0.010 \pm 0.011$ with
world-best precision, where the first uncertainty is statistical and the second
is systematic. The pattern of helicity amplitude magnitudes is found to align
with expectations from quark-helicity conservation in $B$ decays. The ratio of
branching fractions $[\mathcal{B}(B^0 \to D^{*-} D_s^{*+}) \times
\mathcal{B}(D_s^{*+} \to D_s^+ \gamma)]/\mathcal{B}(B^0 \to D^{*-} D_s^+)$ is
measured to be $2.045 \pm 0.022 \pm 0.071$ with world-best precision. In
addition, the first observation of the Cabibbo-suppressed $B_s^0 \to D^{*-}
D_s^+$ decay is made with a significance of seven standard deviations. The
branching fraction ratio $\mathcal{B}(B_s^0 \to D^{*-} D_s^+)/\mathcal{B}(B^0
\to D^{*-} D_s^+)$ is measured to be $0.049 \pm 0.006 \pm 0.003 \pm 0.002$,
where the third uncertainty is due to limited knowledge of the ratio of
fragmentation fractions.
