We report an unusual angular-dependent exchange bias effect in ferromagnet/antiferromagnet bilayers, where both ferromagnet and antiferromagnet are epitaxially grown. Numerical model calculations predict an approximately 45° period for the sign switching of the exchange-bias field, depending on the ratio between magnetocrystalline anisotropy and exchange-coupling constant. The switching of the sign is indicative of a competition between a fourfold magnetocrystalline anisotropy of the ferromagnet and a unidirectional anisotropy field of the exchange coupling. This predicted unusual angular-dependent exchange bias and its magnetization switching process are confirmed by measurements on fully epitaxial Co3FeN/MnN bilayers by longitudinal and transverse magneto-optic Kerr effect magnetometry. These results provide a deeper understanding of the exchange coupling phenomena in fully epitaxial bilayers with tailored materials and open up a complex switching energy landscape engineering by anisotropies.