  We present analytical and numerical progress on black-hole binary spin
precession at second post-Newtonian order using multi-timescale methods. In
addition to the commonly used effective spin which acts as a constant of
motion, we exploit the weighted spin difference and show that such
reparametrization cures the coordinate singularity that affected the previous
formulation for the case of equal-mass binaries. The dynamics on the precession
timescale is written down in closed form in both coprecessing and inertial
frames. Radiation reaction can then be introduced in a quasi-adiabatic fashion
such that, at least for binaries on quasi-circular orbits, gravitational
inspirals reduce to solving a single ordinary differential equation. We provide
a broad review of the resulting phenomenology and rewrite the relevant physics
in terms of the newly adopted parametrization. This includes the spin-orbit
resonances, the up-down instability, spin propagation at past time infinity,
and new precession estimators to be used in gravitational-wave astronomy. Our
findings are implemented in version 2 of the public Python module PRECESSION.
Performing a precession-averaged post-Newtonian evolution from/to arbitrarily
large separation takes $\lesssim 0.1$ s on a single off-the-shelf processor - a
50x speedup compared to our previous implementation. This allows for a wide
variety of applications including propagating gravitational-wave posterior
samples as well as population-synthesis predictions of astrophysical nature.
