You know what a cube is. One of the five Platonic solids, it has six square faces, 12 vertices, and 8 edges
The dual of the cube is the octahedron. As a result of this, the internal mechanism of a face-turning cube puzzle is the same as the internal mechanism of a vertex-turning octahedron puzzle.
Cubes have a total of 24 rotational isometries. You can affirm this for yourself in the following way. Take a Rubik’s Cube and put it down on the table in front of you so that one face is facing directly towards you. You are going to count the number of ways you can do that.
First, pick a colour to be the top face; there are six different ways you can do that. The bottom face is now fixed, but you can choose between the four remaining colours to be the front face. That fixes all the other colours, so the number of isometries is 6 x 4 = 24.
This method works no matter what features you choose. You could start by picking a vertex — there’s eight ways to do that. With a vertex fixed, there are only three edges and three faces to choose from, so the total number of isometries is 8 x 3 = 24.
There are 13 axes of rotation.
Three of them pass through the centre of opposite faces. These are the turning axes of the Rubik’s Cube and other magic cube puzzles. They have 90° symmetry, allowing us to cycle through four isometries.
Another 4 pass through opposite vertices, and have 120° symmetry, so they cycle through three isometries. These are the turning axes of the vertex-turning cube puzzles, such as the Rex Cube and the Mosaic Cube.
The remaining 6 pass through the centre of opposite edges, and have 180° symmetry. Thus they let us swap back and forth between two isometries. These are used as the turning axes of the Helicopter Cube and Curvy Copter puzzles, and as some turning axes of the Fisher’s Cube.
We can be sure there are no other axes of rotation by noting that all our isometries are accounted for. Starting from any chosen isometry (that’s one), we can reach three new isometries by rotating around any one of the three face pair axes (3 x 3 = 9, so now we’re up to ten). We can reach two new isometries by rotating around any of the four vertex-pair axes (2 x 4 = 8, so we’re up to 18). Finally, we can reach a new isometry by rotating 180° around any of the 6 edge-pair axes. (that’s 24!)