  We derive the third subleading (N$^3$LO) corrections of the quadratic-in-spin
sectors via the EFT of spinning objects in post-Newtonian (PN) gravity. These
corrections consist of contributions from $4$ sectors for generic compact
binaries, that enter at the fifth PN order. One of these contributions is due
to a new tidal interaction, that is unique to the sectors with spin, and
complements the first tidal interaction that also enters at this PN order in
the simple point-mass sector. The evaluation of Feynman graphs is carried out
in a generic dimension via advanced multi-loop methods, and gives rise to
dimensional-regularization poles in conjunction with logarithms. At these
higher-spin sectors the reduction of generalized Lagrangians entails
redefinitions of the position beyond linear order. We provide here the most
general Lagrangians and Hamiltonians. We then specify the latter to simplified
configurations, and derive the consequent gauge-invariant relations among the
binding energy, angular momentum, and frequency. We end with a derivation of
all the scattering angles that correspond to an extension of our Hamiltonians
to the scattering problem in the simplified aligned-spins configuration, as a
guide to scattering-amplitudes studies.
