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.
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