  By extending our recent framework to describe the tidal deformations of a
spinning compact object, we compute for the first time the tidal Love numbers
of a spinning neutron star to linear order in the angular momentum. The spin of
the object introduces couplings between electric and magnetic distortions and
new classes of spin-induced ("rotational") tidal Love numbers emerge. We focus
on stationary tidal fields, which induce axisymmetric perturbations. We present
the perturbation equations for both electric-led and magnetic-led rotational
Love numbers for generic multipoles and explicitly solve them for various
tabulated equations of state and for a tidal field with an electric (even
parity) and magnetic (odd parity) component with $\ell=2,3,4$. For a binary
system close to the merger, various components of the tidal field become
relevant. In this case we find that an octupolar magnetic tidal field can
significantly modify the mass quadrupole moment of a neutron star. Preliminary
estimates, assuming a spin parameter $\chi\approx0.05$, show modifications
$\gtrsim10\%$ relative to the static case, at an orbital distance of five
stellar radii. Furthermore, the rotational Love numbers as functions of the
moment of inertia are much more sensitive to the equation of state than in the
static case, where approximate universal relations at the percent level exist.
For a neutron-star binary approaching the merger, we estimate that the
approximate universality of the induced mass quadrupole moment deteriorates
from $1\%$ in the static case to roughly $6\%$ when $\chi\approx0.05$. Our
results suggest that spin-tidal couplings can introduce important corrections
to the gravitational waveforms of spinning neutron-star binaries approaching
the merger.
