At the very superficial level at which I operate, it seems as if counter-rotating turbines, would have one efficiency advantage. Instead of wasting energy trying to rotate the stators, the swirl from one turbine would actually (help) turn the next one. I'm sure the energy lost to the stators is small (though it can't be zero), and that you'd lose much more in other ways if you tried to build counter-rotating turbines.
But am I right that this is a theoretical advantage of counter-rotating turbines? And if so, is there a corresponding advantage to counter-rotating fans (I can't quite seem to get my head around that).