  Recently, the gravitational scattering of two black holes (BHs) treated at
the leading order in the weak-field, or post-Minkowskian (PM), approximation to
General Relativity has been shown to map bijectively onto a simpler effectively
one-body process: the scattering of a test BH in a stationary BH spacetime.
Here, for BH spins aligned with the orbital angular momentum, we propose a
simple extension of that mapping to 2PM order. We provide evidence for the
validity and utility of this 2PM mapping by demonstrating its compatibility
with all known analytical results for the conservative local-in-time dynamics
of binary BHs in the post-Newtonian (weak-field and slow-motion) approximation
and, separately, in the test-BH limit. Our result could be employed in the
construction of improved effective-one-body models for the conservative
dynamics of inspiraling spinning binary BHs.
