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Fully relativistic first-principles calculations of the Fe(001) surface demonstrate that resonant surface (interface) states may produce sizable tunneling anisotropic magnetoresistance in magnetic tunnel junctions with a single magnetic electrode. The effect is driven by the spin-orbit coupling. It shifts the resonant surface band via the Rashba effect when the magnetization direction changes. We find that spin-flip scattering at the interface is controlled not only by the strength of the spin-orbit coupling, but depends strongly on the intrinsic width of the resonant surface states.