Abstract: The first full angular analysis of the B0→D∗−Ds∗+ 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 Ds∗+→Ds+γ and D− → D¯0π− vector meson decays are used with the subsequent Ds+ → K+K−π+ and D¯0 → K+π− decays. All helicity amplitudes and phases are measured, and the longitudinal polarisation fraction is determined to be fL = 0.578 ± 0.010 ± 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 [ℬ(B0→D∗−Ds∗+) × ℬ(Ds∗+→Ds+γ)]/ℬ(B0 → D−Ds+) is measured to be 2.045 ± 0.022 ± 0.071 with world-best precision. In addition, the first observation of the Cabibbo-suppressed Bs → D−Ds+ decay is made with a significance of seven standard deviations. The branching fraction ratio ℬ(Bs → D−Ds+)/ℬ(B0 → D*−Ds+) is measured to be 0.049 ± 0.006 ± 0.003 ± 0.002, where the third uncertainty is due to limited knowledge of the ratio of fragmentation fractions.