Oh oh, I *love* it when cats sleep in strange positions. I've never had a cat that would sleep flat on it's back, but I did have an utterly shameless little lady cat who when picked up by *anybody* would roll over on her back, stretch her paws as far they would go and just hang there waiting to have her belly rubbed...

One of my favourite is when a cat is dozing all dignified in the "Sphinx position", front paws neatly folded in underneath... and then gradually sags forward until his forehead is resting flat on the ground...

And I had a cat who would beg to be let in by jumping up and hanging on the slats across the window. If we didn't let him in immediately he'd climb up another slat, hang there at full stretch and miaow piteously at us through the window. By then we were usually helpless with laughter so it took a while to go and open the door