A bstract The first full angular analysis of the $$ {B}^0\to {D}^{\ast -}{D}_s^{\ast +} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>B</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mo>−</mml:mo> </mml:mrow> </mml:msup> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> 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 $$ {D}_s^{\ast +}\to {D}_s^{+}\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>→</mml:mo> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mi>γ</mml:mi> </mml:math> and D * − → $$ {\overline{D}}^0{\pi}^{-} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msup> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>−</mml:mo> </mml:msup> </mml:math> vector meson decays are used with the subsequent $$ {D}_s^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> → K + K − π + and $$ {\overline{D}}^0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msup> </mml:math> → 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( $$ {B}^0\to {D}^{\ast -}{D}_s^{\ast +} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>B</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mo>−</mml:mo> </mml:mrow> </mml:msup> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> ) × ℬ( $$ {D}_s^{\ast +}\to {D}_s^{+}\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>→</mml:mo> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mi>γ</mml:mi> </mml:math> )] / ℬ( B 0 → D * − $$ {D}_s^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> ) 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 * − $$ {D}_s^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> decay is made with a significance of seven standard deviations. The branching fraction ratio ℬ( B s → D * − $$ {D}_s^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> ) / ℬ( B 0 → D * − $$ {D}_s^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> ) 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.