Understanding of the magnetocrystalline anisotropy in magnetic materials (the influence of different elemental components on the direction of easy magnetization) can be greatly enhanced by measuring the orbital moment anisotropy of the elemental constituents. A circular x-ray dichroism technique is presented that allows the determination of the microscopic origin of the spin reorientation transition in ultrathin single-crystalline cobalt/nickel films, The stronger anisotropy contribution of a much thinner cobalt layer redirects the easy magnetization direction of the entire film.