  A novel approach for extracting gauge-invariant information about spin-orbit
coupling in gravitationally interacting binary systems is introduced. This
approach is based on the "scattering holonomy", i.e. the integration (from the
infinite past to the infinite future) of the differential spin evolution along
the two worldlines of a binary system in hyperboliclike motion. We apply this
approach to the computation, at the first post-Minkowskian approximation (i.e.
first order in $G$ and all orders in $v/c$), of the values of the two
gyrogravitomagnetic ratios describing spin-orbit coupling in the Effective
One-Body formalism. These gyrogravitomagnetic ratios are found to tend to zero
in the ultrarelativistic limit.
