The problem of higher-order Néel anisotropies is solved by exploiting the addition theorem for spherical functions. A key advantage of the present approach is the orthonormal character of the expansion of the magnetic energy that simplifies the formalism and makes possible the treatment of nonideal morphologies as well. Explicit expressions for second-, fourth-, and sixth-order anisotropies are obtained for ideal bulk of fcc and bcc symmetry as well as for (001), (110), and (111) surfaces with nearest-neighbor (NN) Néel interactions. The systematic examination of the pair model involves partition by species of inequivalent sites, interaction spheres, and orders in the multipole expansion. It enables us to to treat also next-nearest-neighbor (NNN) pair interactions to the same high orders as the NN ones. The analysis sheds light onto the peculiar cases of bcc(100) and bcc(111) surfaces where one finds no symmetry breaking (no second-order contributions) with NN interactions only. With the extension to NNN's, it is demonstrated that bcc(111) surfaces exhibit a particularly high symmetry and acquire no second-order anisotropy contributions from NNN interactions, whereas the latter induce a second-order symmetry breaking in the bcc(100) case