It sounds like you've done all the right things: swapping speakers and speaker wires to try to diagnose the problem. The only other thing that I can think of is to inspect the center channel speaker terminals on the 605 and see if anything stands out to you.
Given what you've said, I suspect that you got a defective unit. That's always the last thing that I want to say, but I think it's the case. If you swapped speakers and wires (that you know work), and the auto setup still won't detect a center channel, I don't know what else would cause that.
If I'm missing something, I sure hope that others here will chime in. Otherwise, my advice is to contact Amazon for a replacement. I don't believe that you'll be charged anything at all if the unit is defective.