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The anisotropy of fcc (111) Fe/Ni and Fe/Ni/Fe layers is investigated. The films exhibit an oscillating behavior of the preferential magnetization direction depending on whether the surface layer is Ni or Fe. Anisotropy fields are obtained from polar and perpendicular Kerr hysteresis loops and yield a comparatively small perpendicular Fe/Ni interface anisotropy of about 0.15 mJ/m2. The real-space origin of the interface anisotropy is the interlayer hybridization of the yz, zx, and z2 orbitals.