Turning a video sphere inside out

The real fun in this spherical video starts at 3:43. Use the directional pad in the upper left corner of the YouTube window to center on Emily's face as the room undergoes a Moebius inversion, which turns the sphere inside out.

Many of you will have no trouble understanding the math. I don't understand it, but I watched this video about Moebius transformations and now I can pretend to get what's going on.