We consider a binary system of spinning compact objects with internal structure, moving along an exactly circular orbit, and modeled 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 four-velocity, four-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.