  We study the link between classical scattering of spinning black holes and
quantum amplitudes for massive spin-$s$ particles. Generic spin orientations of
the black holes are considered, allowing their spins to be deflected on par
with their momenta. We rederive the spin-exponentiated structure of the
relevant tree-level amplitude from minimal coupling to Einstein's gravity,
which in the $s\to\infty$ limit generates the black holes' complete series of
spin-induced multipoles. The resulting scattering function is seen to encode in
a simple way the known net changes in the black-hole momenta and spins at first
post-Minkowskian order. We connect our findings to a rigorous framework
developed elsewhere for computing such observables from amplitudes.
