  Many models currently exist which attempt to interpret the excess of gamma
rays emanating from the Galactic Center in terms of annihilating or decaying
dark matter. These models typically exhibit a variety of complicated cascade
mechanisms for photon production, leading to a non-trivial kinematics which
obscures the physics of the underlying dark sector. In this paper, by contrast,
we observe that the spectrum of the gamma-ray excess may actually exhibit an
intriguing "energy-duality" invariance under $E_\gamma \rightarrow
E_\ast^2/E_\gamma$ for some $E_\ast$. As we shall discuss, such an energy
duality points back to a remarkably simple alternative kinematics which in turn
is realized naturally within the Dynamical Dark Matter framework. Observation
of this energy duality could therefore provide considerable information about
the properties of the dark sector from which the Galactic-Center gamma-ray
excess might arise, and highlights the importance of acquiring more complete
data for the Galactic-Center excess in the energy range around 1 GeV.

High Energy Physics - Phenomenology;Cosmology and Nongalactic Astrophysics;Astrophysics of Galaxies;High Energy Astrophysical Phenomena