  The structure of a two-dimensional honeycomb optical lattice potential with
small inversion asymmetry is characterized using coherent diffraction of
$^{87}$Rb atoms. We demonstrate that even a small potential asymmetry, with
peak-to-peak amplitude of $\leq 2.3\%$ of the overall lattice potential, can
lead to pronounced inversion asymmetry in the momentum-space diffraction
pattern. The observed asymmetry is explained quantitatively by considering both
Kapitza-Dirac scattering in the Raman-Nath regime, and also either perturbative
or full-numerical treatment of the band structure of a periodic potential with
a weak inversion-symmetry-breaking term. Our results have relevance for both
the experimental development of coherent atom optics and the proper
interpretation of time-of-flight assays of atomic materials in optical
lattices.
