We investigate the anisotropic nature of magnetocrystalline coupling between the crystallographic and skyrmion crystal (SKX) lattices in the chiral magnet MnSi by magnetic field-angle resolved resonant ultrasound spectroscopy. Abrupt changes are observed in the elastic moduli and attenuation when the magnetic field is parallel to the [011] crystallographic direction. These observations are interpreted in a phenomenological Ginzburg-Landau theory that identifies switching of the SKX orientation to be the result of an anisotropic magnetocrystalline coupling potential. Our paper sheds new light on the nature of magnetocrystalline coupling potential relevant to future spintronic applications.