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Tunneling anisotropic magnetoresistance (TAMR) is observed in tunnel junctions with transition metal electrodes as the moments are rotated from in-plane to out-of-plane in sufficiently large magnetic fields that the moments are nearly parallel to one another. A complex angular dependence of the tunneling resistance is found with twofold and fourfold components that vary strongly with bias voltage. Distinctly different TAMR behaviors are obtained for devices formed with highly textured crystalline MgO(001) and amorphous Al2O3 tunnel barriers. A tight-binding model shows that a fourfold angular dependence can be explained by the presence of an interface resonant state that affects the transmission of the contributing tunneling states through a spin-orbit interaction.