  We compute the first-order self-force contribution to Detweiler's redshift
invariant for extended bodies endowed with both dipolar and quadrupolar
structure (with spin-induced quadrupole moment) moving along circular orbits on
a Schwarzschild background. Our analysis includes effects which are second
order in spin, generalizing previous results for purely spinning particles. The
perturbing body is assumed to move on the equatorial plane, the associated spin
vector being orthogonal to it. The metric perturbations are obtained by using a
standard gravitational self-force approach in a radiation gauge. Our results
are accurate through the 6.5 post-Newtonian order, and are shown to reproduce
the corresponding post-Newtonian expression for the same quantity computed by
using the available Hamiltonian from an effective field theory approach for the
dynamics of spinning binaries.
