A family of models of partially relaxed stellar systems I. Dynamical properties

[Abridged] Recently we have found that a family of models of partially relaxed, anisotropic stellar systems, inspired earlier by studies of incomplete violent relaxation, exhibits some interesting thermodynamic properties. Here we present a systematic investigation of its dynamical characteristics, in order to establish the basis for a detailed comparison with simulations of collisionless collapse, planned for a separate paper. For a full comparison with the observations of elliptical galaxies, the models should be extended to allow for the presence a sizable dark halo and of significant rotation. In the spherical limit, the family is characterized by two dimensionless parameters, i.e. \Psi, measuring the depth of the galaxy potential, and \nu, defining the form of a third global quantity Q, which is argued to be approximately conserved during collisionless collapse (in addition to the total energy and the total number of stars)