We demonstrate the occurrence of tunneling anisotropic magnetoresistance on the atomic scale using scanning tunneling microscopy (STM). Our experiments show that noncollinear atomic-scale magnetic structures can be revealed in STM using nonmagnetic tips. These observations can be explained by a variation of the local density of states of an atom depending on its magnetization direction due to spin-orbit interaction. STM simulations based on this effect are in excellent agreement with the experimental images and can explain the bias-voltage-dependent contrast found for two-dimensionally modulated spin textures.