  We derive an effective potential for binary black-hole (BBH) spin precession
at second post-Newtonian order. This effective potential allows us to solve the
orbit-averaged spin-precession equations analytically for arbitrary mass ratios
and spins. These solutions are quasiperiodic functions of time: after a fixed
period the BBH spins return to their initial relative orientations and jointly
precess about the total angular momentum by a fixed angle. Using these
solutions, we classify BBH spin precession into three distinct morphologies
between which BBHs can transition during their inspiral. We also derive a
precession-averaged evolution equation for the total angular momentum that can
be integrated on the radiation-reaction time and identify a new class of
spin-orbit resonances that can tilt the direction of the total angular momentum
during the inspiral. Our new results will help efforts to model and interpret
gravitational waves from generic BBH mergers and predict the distributions of
final spins and gravitational recoils.
