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The magnetization reversal in exchange-biased ferromagnetic-antiferromagnetic (FM-AFM) bilayers is investigated. Different reversal pathways on each branch of the hysteresis loop, ie, asymmetry, are obtained both experimentally and theoretically when the magnetic field is applied at certain angles from the anisotropy direction. The range of angles and the magnitude of this asymmetry are determined by the ratio between the FM anisotropy and the interfacial FM-AFM exchange anisotropy. The occurrence of asymmetry is linked with the appearance of irreversibility, ie, finite coercivity, as well as with the maximum of exchange bias, increasing for larger anisotropy ratios. Our results indicate that asymmetric hysteresis loops are intrinsic to exchange-biased systems and the competition between anisotropies determines the asymmetric behavior of the magnetization reversal.