  The quadruple perovskite CaMn$_7$O$_{12}$ is a topical multiferroic, in which
the hierarchy of electronic correlations driving structural distortions,
modulated magnetism, and orbital order is not well known and may vary with
temperature. x-ray resonant elastic scattering (XRES) provides a
momentum-resolved tool to study these phenomena, even in very small single
crystals, with valuable information encoded in its polarization- and
energy-dependence. We present an application of this technique to
CaMn$_7$O$_{12}$. By polarization analysis, it is possible to distinguish
superstructure reflections associated with magnetic order and orbital order.
Given the high momentum resolution, we resolve a previously unknown splitting
of an orbital order superstructure peak, associated with a distinct
\textit{locked-in} phase at low temperatures. A second set of orbital order
superstructure peaks can then be interpreted as a second-harmonic orbital
signal. Surprisingly, the intensities of the first- and second-harmonic orbital
signal show disparate temperature and polarization dependence. This orbital
re-ordering may be driven by an exchange mechanism, that becomes dominant over
the Jahn-Teller instability at low temperature.
