The interplay between magnetocrystalline anisotropy and exchange bias is investigated in CoO/Co bilayer films, which are grown epitaxially on MgO (0 0 1), by magnetization reversal measurements based on the anisotropic magnetoresistance (AMR) effect. While an asymmetric magnetization reversal survives after training for cooling field (CF) along the hard axis, the magnetization reversal becomes symmetric and is dominated in both branches of the hysteresis loop by domain wall motion before and after training for CF along the easy axis. When performing an in-plane hysteresis loop perpendicular to the CF, the hysteresis loop along the easy axis becomes asymmetric: magnetization rotation dominates in the ascending branch, while there is a larger contribution of domain wall motion in the descending branch. Furthermore, the azimuthal angular dependence of the AMR shows two minima after performing a perpendicular hysteresis loop, instead of only one minimum after training. Relying on the extended Fulcomer and Charap model, these effects can be related to an increased deviation of the average uncompensated antiferromagnetic magnetization from the CF direction. This model provides a consistent interpretation of training and asymmetry of the magnetization reversal for epitaxial films with pronounced magnetocrystalline anisotropy as well as for the previously investigated polycrystalline films.