Mark, the paper of mine you're remembering is the one joint with Peter May, "Algebras over equivariant sphere spectra", JPAA 116 (1997) 139-149. It shows that, indeed, you can consider just one underlying category of G-spectra with a number of (cofibrantly generated) model structures on it. The "change of universe" functors usually considered from the Lewis-May-Steinberger point of view are then simply the derived functors of the identity functor. The category itself is the EKMM category with G acting in the obvious way on the objects. -- Tony Elmendorf