The only exercise mat I have seen for the NES didn't have anything on the other side of it. Of course, I have only seen one, about 20+ years ago.
The 'ring' thing is the Activator that Sega released for the Genesis (EGM did several specials on it around issue 30 to 45, not sure which one specifically but that is the range I think it is in). It was interesting, just ahead of it's time, same thing with the Power Glove (why hasn't someone just made a pair of gloves you could strap the Wiimote to and viola!, Power Glove).