The first full angular analysis of the B0→D∗−Ds∗+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {B}^0\to {D}^{\ast -}{D}_s^{\ast +} $$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{\ast +}\to {D}_s^{+}\gamma $$\end{document} and D^*− → D¯0π−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{D}}^0{\pi}^{-} $$\end{document} vector meson decays are used with the subsequent Ds+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{+} $$\end{document} → K⁺K⁻π⁺ and D¯0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{D}}^0 $$\end{document} → K⁺π⁻ decays. All helicity amplitudes and phases are measured, and the longitudinal polarisation fraction is determined to be f_L = 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∗+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {B}^0\to {D}^{\ast -}{D}_s^{\ast +} $$\end{document}) × ℬ(Ds∗+→Ds+γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{\ast +}\to {D}_s^{+}\gamma $$\end{document})]/ℬ(B⁰ → D^*−Ds+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{+} $$\end{document}) is measured to be 2.045 ± 0.022 ± 0.071 with world-best precision. In addition, the first observation of the Cabibbo-suppressed B_s → D^*−Ds+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{+} $$\end{document} decay is made with a significance of seven standard deviations. The branching fraction ratio ℬ(B_s → D^*−Ds+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{+} $$\end{document})/ℬ(B⁰ → D^*−Ds+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{+} $$\end{document}) 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.