  We resolve subtleties in calculating the post-Minksowskian dynamics of binary
systems, as a spin expansion, from massive scattering amplitudes of fixed
finite spin. In particular, the apparently ambiguous spin Casimir terms can be
fully determined from the gradient of the spin-diagonal part of the amplitudes
with respect to $S^2 = -s(s+1)\hbar^2$, using an interpolation between massive
amplitudes with different spin representations. From two-loop amplitudes of
spin-0 and spin-1 particles minimally coupled to gravity, we extract the spin
Casimir terms in the conservative scattering angle between a spinless and a
spinning black hole at $\mathcal{O}(G^3 S^2)$, finding agreement with known
results in the literature. This completes an earlier study [Phys. Rev. Lett.
130 (2023), 021601] that calculated the non-Casimir terms from amplitudes. We
also illustrate our methods using a model of spinning bodies in
electrodynamics, finding agreement between scattering amplitude predictions and
classical predictions in a root-Kerr electromagnetic background up to
$\mathcal{O}(\alpha^3 S^2)$. For both gravity and electrodynamics, the finite
part of the amplitude coincides with the two-body radial action in the aligned
spin limit, generalizing the amplitude-action relation beyond the spinless
case. Surprisingly, the two-loop amplitude displays a hidden spin-shift
symmetry in the probe limit, which was previously observed at one loop. We
conjecture that the symmetry holds to all orders in the coupling constant and
is a consequence of integrability of Kerr orbits in the probe limit at the
first few orders in spin.
