In this paper we confirm the generalized actions of the complete NLO
cubic-in-spin interactions for generic compact objects which were tackled first
via an extension of the EFT of spinning gravitating objects. The interaction
potentials are made up of $6$ independent sectors, including a new unique
sector that is proportional to the square of the quadrupolar deformation
parameter, $C_{ES^2}$. We derived the full Hamiltonians in an arbitrary
reference frame and in generic kinematic configurations. Using these most
general Hamiltonians we find the full Poincar\’e algebra of all the sectors at
the 4.5PN order, including the third subleading spin-orbit sector recently
derived within our approach. We also derive the binding energies with
gauge-invariant relations useful for gravitational-wave applications. Finally,
we derive the extrapolated scattering angles in the aligned-spins case, and we
find complete agreement with previous results derived for the scattering of
black holes via scattering-amplitudes methods. The completion of the full
Poincar\’e algebra at the 4.5PN order provides a strong validation that this
new precision frontier in PN theory has now been established.
