  We confirm the generalized actions of the complete NLO cubic-in-spin
interactions for generic compact binaries which were first tackled via an
extension of the EFT of spinning gravitating objects. We first reduce these
generalized actions to standard actions with spins, where the interaction
potentials are found to consist of $6$ independent sectors, including a new
unique sector that is proportional to the square of the quadrupolar deformation
parameter, $C_{\text{ES}^2}$. We derive the general Hamiltonians in an
arbitrary reference frame, and for generic kinematic configurations. With these
most general Hamiltonians we construct the full Poincar\'e algebra of all the
sectors at the fourth and a half post-Newtonian (4.5PN) order, including the
third subleading spin-orbit sector, recently accomplished uniquely via our
framework, thus proving the Poincar\'e invariance of all relevant sectors. We
then derive the binding energies with gauge-invariant relations useful for
gravitational-wave applications. Finally, we also derive the extrapolated
scattering angles in the aligned-spins configuration for the scattering
problem. Yet, as made clear already as of quadratic-in-spin sectors, the
aligned-spins simplification inherent to the scattering-angle observable,
entails a great loss of physical information, that is only growing with
higher-spin sectors. Our completion of the full Poincar\'e algebra at the
present $4.5$PN order provides strong confidence that this new precision
frontier in PN theory has now been established.
