  The relative equilibria of planar Newtonian $N$-body problem become coorbital
around a central mass in the limit when all but one of the masses becomes zero.
We prove a variety of results about the coorbital relative equilibria, with an
emphasis on the relation between symmetries of the configurations and
symmetries in the masses, or lack thereof. We prove that in the $N=4$, $N=6$,
and $N=8$ Newtonian coorbital problems there exist symmetric relative
equilibria with asymmetric positive masses. This result can be generalized to
other homogeneous potentials, and we conjecture similar results hold for larger
even numbers of infinitesimal masses. We prove that some equalities of the
masses in the $1+4$ and $1+5$ coorbital problems imply symmetry of a class of
convex relative equilibria. We also prove there is at most one convex central
configuration of the symmetric $1+5$ problem.
