A generalization of the Osipkov-Merritt inversion of the fundamental integral equation for anisotropic spherical galaxies is obtained. For any density profile this inversion yields a two-parameter distribution function with arbitrary degrees of both radial and tangential velocity anisotropies. Therefore, for the first time, exact analytical models can be found which generate given density profiles and possess tangential velocity anisotropies. This property disposes of the well-known radial orbit instability encountered by many models of anisotropic spherical galaxies. Moreover, the inversion provides a new opportunity to study instabilities in analytical galaxy models which have realistic density profiles. As examples of the inversion, two families of anisotropic Plummer and Jaffe models are calculated. These models are completely analytical in that the distribution functions, velocity dispersion profiles and differential energy distributions can all be expressed in terms of elementary functions. Some interesting qualitative properties of these models are also discussed.