We study the anisotropy of the spin and orbital magnetization of ultrathin Co layers characterized by structure inversion asymmetry, namely Pt/Co/AlO${}_{x}$ trilayers with Co thicknesses between 0.6 and 2 nm. We use x-ray magnetic circular dichroism (XMCD) to probe the Co layer along two orthogonal measurement geometries. By combining transverse and collinear XMCD, we achieve a vector measurement of the Co spin and orbital magnetic moments. We find an enhanced and anisotropic orbital magnetic moment localized at the Co interface and verify the connection between the Co orbital moment anisotropy and the macroscopic magnetic anisotropy of the trilayer. These results provide a link between diverse interfacial phenomena originating from spin-orbit coupling, such as the Rashba effect and perpendicular magnetic anisotropy. Using simple considerations derived from tight-binding models, we show that the Rashba constant is proportional to the ratio between out-of-plane and in-plane orbital magnetic moments.