Cross-sectional scanning tunneling microscopy is used to identify the origin of the anisotropic electronic structure of acceptor states in III-V semiconductors. The density of states introduced by a hole bound to an individual Cd acceptor in GaP is spatially mapped at room temperature. Similar to the Mn hole wave function in GaAs, we found a highly anisotropic, crosslike shape of the hole bound to Cd both at the GaP(110) and the GaP(1 (1) over bar0) orthogonal cleavage planes. The experimentally observed similarity of the symmetry properties of Mn:GaAs to Cd:GaP shows that the anisotropic structure of acceptor states in zinc-blende III-V compounds is determined by the cubic symmetry of the host crystal. Nevertheless, the weak spin-orbit int...