  Tidal forces cause inspiralling binary neutron stars to deform, leaving a
measurable imprint on the gravitational waves they emit. The induced stellar
multipoles are an added source of gravitational radiation and modify the
orbital dynamics, producing a slight acceleration of the coalescence which
manifests as a phase shift in the waveform relative to point-particles. The
dominant piece of this tidal phase comes from the mass quadrupoles, which
contribute at fifth post-Newtonian order (5PN). Current quadrupoles and mass
octupoles contribute at higher orders. For spinning neutron stars, additional
multipole moments are induced by nonlinear couplings between spin and tides. We
calculate these rotational-tidal deformations assuming the stars are rotating
slowly and the tides are weak and quasi-stationary. The stellar multipole
moments are read off from an asymptotically flat metric that encodes the
difference between their tidal response and a black hole's. The multipoles are
subsequently inserted into post-Newtonian formulas for the orbit and the
gravitational radiation. We find that, at leading order, the rotational-tidal
deformations make a 6.5PN contribution to the tidal phase. Their effect on the
waveform is thus larger than that of the mass octupoles, and nearly as large as
that of the current quadrupoles, in systems with non-negligible spin.
