Magnetic anisotropies of molecular-beam-epitaxy-grown fcc Co(110) films on Cu(110) single-crystal substrates have been determined by using Brillouin light scattering and have been correlated with the structural properties determined by low-energy electron diffraction and scanning tunneling microscopy (STM). Three regimes of film growth and associated anisotropy behavior are identified: coherent growth in the Co film thickness regime of up to 13 Å, in-plane anisotropic strain relaxation between 13 and about 50 Å and in-plane isotropic strain relaxation above 50 Å. The structural origin of the transition between anisotropic and isotropic strain relaxation was studied using STM. In the regime of anisotropic strain relaxation long Co stripes with a preferential [11¯0]-orientation are observed, which in the isotropic strain relaxation regime are interrupted in the perpendicular in-plane direction to form isotropic islands. In the Co film thickness regime below 50 Å an unexpected suppression of the magnetocrystalline anisotropy contribution is observed. Symmetry reflections based on a crystal-field formalism and discussed within the context of band theory, which explicitly takes tetragonal misfit strains into account, reproduce the experimentally observed anomalies despite the fact that the thick Co films are quite rough.