The point of this puzzle is to position the pyramids inside the sphere in such a way that the two colors that share an edge are the same color. The pieces are moved by rotating them into the empty space.
In this solve, the puzzle starts in one solved state and is manipulated to get to the other solved state. In the starting state of the puzzle, blue is on the left if red is placed on the bottom. After the solve, blue is on the right when red is on the bottom. This clearly shows that the puzzle has changed from one solution to the other.
In order to accomplish this, a "dexterity move" is required, which involves rotating a single pyramid 90 degrees, a move that is impossible with normal mathematical moves. That is what I am doing in the beginning of the solve.