  We obtain the quadratic-in-spin terms of the conservative Hamiltonian
describing the interactions of a binary of spinning bodies in General
Relativity through $\mathcal{O}(G^2)$ and to all orders in velocity. Our
calculation extends a recently-introduced framework based on scattering
amplitudes and effective field theory to consider non-minimal coupling of the
spinning objects to gravity. At the order that we consider, we establish the
validity of the formula proposed in \cite{Bern:2020buy} that relates the
impulse and spin kick in a scattering event to the eikonal phase.
