  We demonstrate equivalences, under simple mappings, between the dynamics of
three distinct systems---(i) an arbitrary-mass-ratio two-spinning-black-hole
system, (ii) a spinning test black hole in a background Kerr spacetime, and
(iii) geodesic motion in Kerr---when each is considered in the first
post-Minkowskian (1PM) approximation to general relativity, i.e. to linear
order $G$ but to all orders in $1/c$, and to all orders in the black holes'
spins, with all orders in the multipole expansions of their linearized
gravitational fields. This is accomplished via computations of the net results
of weak gravitational scattering encounters between two spinning black holes,
namely the net $O(G)$ changes in the holes' momenta and spins as functions of
the incoming state. The results are given in remarkably simple closed forms,
found by solving effective Mathisson-Papapetrou-Dixon-type equations of motion
for a spinning black hole in conjunction with the linearized Einstein equation,
with appropriate matching to the Kerr solution. The scattering results fully
encode the gauge-invariant content of a canonical Hamiltonian governing
binary-black-hole dynamics at 1PM order, for generic (unbound and bound) orbits
and spin orientations. We deduce one such Hamiltonian, which reproduces and
resums the 1PM parts of all such previous post-Newtonian results, and which
directly manifests the equivalences with the test-body limits via simple
effective-one-body mappings.
