We report a detailed study of the magnetic properties of the site-random solid solution Fe1-xCoxCl2. Fe1-xCoxCl2 represents an archetypal example of a system with competing orthogonal spin anisotropies; the easy axis of the Fe spin is orthogonal to the easy plane of the Co spin. The magnetic behavior of single-crystal samples was characterized with the use, as probes, of both dc magnetic susceptibility and elastic neutron diffraction. We find behavior that is in fundamental disagreement with the theoretical predictions for random anisotropy magnets. In particular the transition in one spin component is drastically altered by the existence of long-range order in the other component. We argue that this is due to random off-diagonal terms in the Hamiltonian which generate site-random molecular fields. © 1983 The American Physical Society.