Epitaxial FeGa/IrMn bilayers with exchange biases along the FeGa100 and 110 directions are prepared on MgO(001) single crystal substrates by magnetron sputtering through controlling the orientation of the external field in situ applied during growth. The effect of the exchange bias orientation on the magnetic switching process and the magnetic switching field are studied. The X-ray φ-scan indicates that the FeGa layer is epitaxially grown with a 45° in-plane rotation on the MgO(001) substrate along the FeGa(001)110 direction and the MgO(001)100 direction. The measurements of the angular dependence of the ferromagnetic resonance field and the corresponding fitting to the Kittel equation show that the samples have a superposition of fourfold symmetric magnetocrystalline anisotropy K_1 , unidirectional magnetic exchange bias anisotropy K_\mathrme\mathrmb , and uniaxial magnetic anisotropy K_\mathrmu with configuration of K_\mathrme\mathrmb//\left100\right or K_\mathrme\mathrmb//\left110\right . The combined longitudinal and transverse magneto-optical Kerr effect measurements show that sample with K_\mathrme\mathrmb//\left100\right exhibits square loops, asymmetrically shaped loops, and one-sided two-step loops in different external magnetic field directions. In contrast, the sample with K_\mathrme\mathrmb//\left110\right exhibits one-sided two-step and two-sided two-step loops as the magnetic field orientation changes. Because the K_1 is superimposed by K_\mathrmu and K_\mathrme\mathrmb , the in-plane fourfold symmetry of the magnetic anisotropy energy is broken. The local minima are no longer strictly along the in-plane \left\langle100\right\rangle directions, but make a deviation angle which depends on the relative orientation and strength of magnetic anisotropy. A model based on the domain wall nucleation and propagation is proposed with considering the different orientations of K_\mathrme\mathrmb , which can nicely explain the change of the magnetic switching route with the magnetic field orientation and fit the angular dependence of the magnetic switching fields, indicating a significant change of domain wall nucleation energy as the orientation of K_\mathrme\mathrmb changes.