  We obtain for the first time all quadratic-in-spin interactions in spinning
binaries at the third subleading order in post-Newtonian (PN) gravity, and
provide their observable binding energies and their gauge-invariant relations
to the angular momentum. Our results are valid for generic compact objects,
orbits, and spin orientations, and enter at the fifth PN order for
maximally-rotating objects, thus pushing the state of the art. This is
accomplished through an extension of the effective field theory of spinning
gravitating objects, and of its computational application. We also discover a
new finite-size effect which is unique to spinning objects, with a new ``Spin
Love number'' as its characteristic coefficient, that is a new probe for
gravity and QCD.
