  Using the spinning worldline quantum field theory formalism we calculate the
quadratic-in-spin momentum impulse $\Delta p_i^\mu$ and spin kick $\Delta
a_i^\mu$ from a scattering of two arbitrarily oriented spinning massive bodies
(black holes or neutron stars) in a weak gravitational background up to third
post-Minkowskian (PM) order ($G^3$). Two-loop Feynman integrals are performed
in the potential region, yielding conservative results. For spins aligned to
the orbital angular momentum we find a conservative scattering angle that is
fully consistent with state-of-the-art post-Newtonian results. Using the 2PM
radiated angular momentum previously obtained by Plefka, Steinhoff and the
present authors we generalize the angle to include radiation-reaction effects,
in which case it avoids divergences in the high-energy limit.
