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We use effective field theory techniques to compute the potentials due tospin-spin and spin-orbit effects, from which the spin(1)spin(2) contribution tothe motion of spinning compact binaries to third Post-Newtonian (PN) orderfollow. We use a formalism which allows us to impose the spin supplementaritycondition (SSC) in a canonical framework to all orders in the PN expansion. Weexplicitly show the equivalence with our previous results, obtained using theNewton-Wigner SSC at the level of the action for spin-spin and spin-orbitpotentials reported in arXiv:gr-qc/0604099 and arXiv:0712.2032[gr-qc]respectively.Comment: 31 pages, 3 figures. v4: All the irrelevant (spin-independent) terms  in eqs (22) and (23) are now omitted, as in eqs (24) and (25), to avoid any  possible confusion regarding the self-consistency of the tetrad ('ee=g'). The  procedure and results in the paper are unaltere

Spin(1)spin(2) effects in the motion of inspiralling compact binaries at third order in the post-Newtonian expansion