Total rotation angle of Newtonian hip-hop orbits

Determine the total rotation angle about the $z$-axis over one period for a given hip-hop symmetric orbit in the Newtonian $(1+2n)$-body problem.

Background

The paper proves the existence of infinitely many hip-hop symmetric periodic orbits on every compact regular energy surface for the Newtonian potential. In the unreduced problem, however, a relative periodic orbit includes rotation about the $z$-axis, and the accumulated rotation angle over one period is relevant for describing the orbit before symmetry reduction. The authors state that their existence result does not provide this angle for any individual orbit.