I did the test, and the outcome is different than I imagined.
Instead of a tube, I used a pair of pants, legs pinned together to give me a more convenient ratio of hole size to torus size. Reaching through one of the legs, I grabbed the other leg and pulled both legs all the way through the hole. I now had two legs inside of each other, with the hole still in the same place.
Two legs inside of each other -- that's a torus, and it's inside out. So as a matter of topology, where form doesn't matter as long as you don't change the way things are connected, the answer to the original question is yes. You can turn a torus inside out, and it's still a torus. But as a matter of geometry, the two are not the same. The two radiuses that characterize the torus switch places. The diameter of torus 1 becomes the thickness of torus 2, and the thickness of torus 1 becomes the diameter of torus 2. So Sozobe was right. The two are part the same, part not the same.
