  We investigate the dynamics of spinning binaries of compact objects at the
next-to-leading order in the quadratic-in-spin effects, which corresponds to
the third post-Newtonian order (3PN). Using a Dixon-type multipolar formalism
for spinning point particles endowed with spin-induced quadrupoles and
computing iteratively in harmonic coordinates the relevant pieces of the PN
metric within the near zone, we derive the post-Newtonian equations of motion
as well as the equations of spin precession. We find full equivalence with
available results. We then focus on the far-zone field produced by those
systems and obtain the previously unknown 3PN spin contributions to the
gravitational-wave energy flux by means of the multipolar post-Minkowskian
(MPM) wave generation formalism. Our results are presented in the
center-of-mass frame for generic orbits, before being further specialized to
the case of spin-aligned, circular orbits. We derive the orbital phase of the
binary based on the energy balance equation and briefly discuss the relevance
of the new terms.
