  Using the gravitational potential and source multipole moments bilinear in
the spins, first computed to next-to-leading order (NLO) in the post-Newtonian
(PN) expansion within the effective field theory (EFT) framework, we complete
here the derivation of the dynamical invariants and flux-balance equations,
including energy and angular momentum. We use these results to calculate
spin-spin effects in the orbital frequency and accumulated phase to NLO for
circular orbits. We also derive the linear momentum and center-of-mass fluxes
and associated kick-velocity, to the highest relevant PN order. We explicitly
demonstrate the equivalence between the quadratic-in-spin source multipoles
obtained using the EFT formalism and those rederived later with more
traditional tools, leading to perfect agreement for spin-spin radiative
observables to NLO among both approaches.
