Relativity and Saving the Phenomena

This is responding to and building on Mike's post about saving the phenomena. I think I would go ahead and expand what Mike said because in general, the whole theory of relativity does seem a little like the saving the phenomena position. On pp. 211-212 of the Einstein-Infeld book, they talk briefly about the fact that you can say either "the sun moves and the earth is at rest" or "the earth moves and the sun is at rest" and that the two statements just signify different frames of reference. They say that the greatness of Copernicus's move can only be understood through the physical point of view. But as for relativity, it does seem to open up several equally acceptable explanations for various observations. I think the big difference is that relativity is supposed to be reality. On p. 212, Einstein and Infeld say, "Could we build a real relativistic physics valid in all CS; a physics in which there would be no place for absolute, but only for relative motion? This is indeed possible!" On p. 213, they say, "Unexpected adventures still await us. But our final aim is always a better understanding of reality." So yeah, it seems to me (and I hardly need mention that I could be misunderstanding relativity) that there are several different explanations, kind of like saving the phenomena. But now that is reality. Whereas the save the phenomena mentality accepts multiple explanations on a mathematical basis, relativity accepts multiple explanations on a physical, real basis. This is quite an "unexpected adventure" and a wild turn in the labyrinth!