We demonstrate that the spin Hall magnetoresistance (SMR) of epitaxial Pt/Co bilayers on MgO(110) single crystal substrates is strongly anisotropic and depends on the applied current direction with respect to the two primary in-plane crystal directions [001] and [1 (1) over bar0], in the Pt layer. SMR results at different temperatures qualitatively suggest the D'yakonov-Perel spin relaxation mechanism through the invariance of the spin diffusion length.s f for a given current direction, while also suggesting an anisotropic Rashba-Edelstein effect from the ratio of lambda ([001])(sf) to lambda([1 (1) over bar0])(sf). Finally, deviation from the standard SMR model spurs the need for new theory that properly characterizes and quantifies the anisotropies in such epitaxial systems.