  We study the process, within classical general relativity, in which an
incident gravitational plane wave, of weak amplitude and long wavelength,
scatters off a massive spinning compact object, such as a black hole or neutron
star. The amplitude of the asymptotic scattered wave, considered here at linear
order in Newton's constant $G$ while at higher orders in the object's multipole
expansion, is a valuable characterization of the response of the object to
external gravitational fields. This amplitude coincides with a classical
($\hbar\to0$) limit of a quantum 4-point (object and graviton in, object and
graviton out) gravitational Compton amplitude, at the tree (linear-in-$G$)
level. Such tree-level Compton amplitudes are key building blocks in
generalized-unitary-based approaches to the post-Minkowskian dynamics of
binaries of spinning compact objects. In this paper, we compute the classical
amplitude using an effective worldline theory to describe the compact object,
determined by an action functional for translational and rotational degrees of
freedom, including couplings of spin-induced higher multipole moments to
spacetime curvature. We work here up to the levels of quadratic-in-spin
quadrupole and cubic-in-spin octupole couplings, respectively involving Wilson
coefficients $C_2$ and $C_3$. For the special case $C_2=C_3=1$ corresponding to
a black hole, we find agreement through cubic-in-spin order between our
classical amplitude and previous conjectures arising from considerations of
quantum scattering amplitudes. We also present new results for general $C_2$
and $C_3$, anticipating instructive comparisons with results from effective
quantum theories.
