  We develop a general formalism for computing classical observables for
relativistic scattering of spinning particles, directly from on-shell
amplitudes. We then apply this formalism to minimally coupled Einstein-gravity
amplitudes for the scattering of massive spin 1/2 and spin 1 particles with a
massive scalar, constructed using the double copy. In doing so we reproduce
recent results at first post-Minkowskian order for the scattering of spinning
black holes, through quadrupolar order in the spin-multipole expansion.
