A week ago we brought home a new kitten. He is very comfortable here and does not seem stressed, just very lovey and playful. I can't get the previous owner to tell me whether or not kitten was having box problems when she had him, so no background there. Not sure of his age except that he's under a year; he was neutered before we got him. Photos here.

I've researched this, (and have 3 previous cats worth of experience), but I can't think of any reason why he is dividing his bowel movements in and out of the box. He does not appear to be in any pain when he eliminates, but for whatever ungodly reason, he keeps peeing in the box then jumping out and pooping near it. He also doesn't bury his pee, he digs to the bottom, pees, and leaves it uncovered. Initially we had too much sand in the box and he used a rug once, so I took most of the sand out, left half completely uncovered and he started peeing in the box again. I'm rewarding him for this, and he's getting that picture. But still not pooping in the box!!! I've tried picking up the waste and putting in the box, then showing it to him. He just sniffs it for a moment, then walks away.

I'm fairly sure this is behavioral, not health related, but I just can not understand why or how to fix it.