  We show that physical scattering observables for compact spinning objects in
general relativity can depend on additional degrees of freedom in the spin
tensor beyond those described by the spin vector alone. The impulse, spin kick,
and leading-order waveforms exhibit such a nontrivial dependence. A signal of
this additional structure is the change in the magnitude of the spin vector
under conservative Hamiltonian evolution, similar to our previous studies in
electrodynamics. These additional degrees of freedom describe dynamical mass
multipoles of compact objects and decouple for black holes. We also show that
the conservative impulse, spin kick and change of the additional degrees of
freedom are encoded in the eikonal phase.
