Well, funny story.
My friend grew irritated with the dern plastic thing, so he threw it at a wall. And guess what? It fell apart.
Inside, the disassembly instructions read:
HOW TO DISASSEMBLE:
To disassemble, throw the ball at the ground.
I'm not kidding.
There are two things that will cause this puzzle to break apart: the first is centripetal force, and the second is significant shock. If every single piece is pulled apart at the same time, then they will all slide away. But, if you pull on just one piece (particularly with tangential force), then they lock up. Also, if you throw it on the ground, the pieces really aren't that well-connected, so they'll crumble away.
The re-assembly is actually pretty intuitive. There are two pieces that are shaped to allow for reassembly, that have an indent in them:
Set these aside. The rest are shaped like this:
Three of these are needed initially, then the two you set aside are needed, then the last piece to go in is one of the 'regular' pieces.
For the rest of this answer, I will be referencing the following example:
Place one piece on the bottom (orange), and hold two pieces vertically adjacent to it (white and green), resting against the first at 90 degree angles. You will have to hold them there with one hand.
Then, pick up each of the pieces set aside (purple and yellow) and place them against the upright pieces horizontally, indented side up. The reason you want to do this is so that the last piece can slide in easily - it's pretty clear that there's more room for the last piece to slide in from the image.
Once these pieces are in, you should be able to hold together easier with just a little bit of pressure on the sides. If you're delicate, you can even get it to sit like this without any interference!
Then, it's as simple as sliding the last piece in, over the indented pieces. It will take a little bit of pressure and bending, but it will snap into place. I find it helps if you pull up on the piece from the other side to get it over the lip. Then boom, puzzle solved.