A general discussion of magnetization in exchange-biased bilayers with higher-order anisotropy contributions in an applied magnetic field is given. Allowed magnetic configurations for a ferromagnet in an applied magnetic field are shown in magnetic phase diagrams that illustrate competing effects of unidirectional, uniaxial, and fourfold anisotropies. The role and asymmetry of energy barriers during magnetization reversal are discussed in detail. A direct comparison with recent experimental results in epitaxial NiFe/FeMn bilayers is provided, and the influence of thermal activation on the magnetization reversal process is discussed.