  For a stationary, axisymmetric, asymptotically flat, ultra-compact [$i.e.$
containing light-rings (LRs)] object, with a $\mathbb{Z}_2$ north-south
symmetry fixing an equatorial plane, we establish that the structure of
timelike circular orbits (TCOs) in the vicinity of the equatorial LRs, for
either rotation direction, depends exclusively on the radial stability of the
LRs. Thus, an unstable LR delimits a region of unstable TCOs (no TCOs) radially
above (below) it; a stable LR delimits a region of stable TCOs (no TCOs)
radially below (above) it. Corollaries are discussed for both horizonless
ultra-compact objects and black holes. We illustrate these results with a
variety of exotic stars examples and non-Kerr black holes, for which we also
compute the efficiency associated with converting gravitational energy into
radiation by a material particle falling under an adiabatic sequence of TCOs.
For most objects studied, it is possible to obtain efficiencies larger than the
maximal efficiency of Kerr black holes, $i.e.$ larger than $42\%$.
