We calculate the spin-orbit corrections to the loss of angular momentum in a two-body scattering at third post-Minkowskian order, O(G3), from scattering amplitudes using the eikonal operator. These results include effects linear in spin, are valid for generic spin orientations, and are presented in a manifestly Poincaré covariant way. We include both radiative losses, by means of the leading-order gravitational waveform, and static losses, by means of the appropriate -i0 prescription in the leading soft graviton theorem, finding agreement with known results in the post-Newtonian limit.