  We consider a binary system of spinning compact objects with internal
structure, moving along an exactly circular orbit, and modelled within the
multipolar gravitational skeleton formalism, up to quadrupolar order. We prove
that the worldline of each multipolar particle is an integral curve of the
helical Killing vector field, and that the 4-velocity, 4-momentum, spin tensor
and quadrupole tensor of each particle are Lie-dragged along those worldlines.
The geometrical framework developed in this paper paves the way to an extension
of the first law of compact-object binary mechanics up to quadrupolar order.
