Apply one of these rotations, then another, and obtain a third rotation in our set of 24. For instance, rotate the cube 90° about the x axis, then 90° about the z axis, and the result is another rigid rotation, accomplished by spinning the cube 120° about its diagonal. Furthermore, each rotation can be "undone" by applying the opposite rotation. Hence the rotations of the cube form a group. If you don't know anything about groups, start here.
The rotations of a solid form a nonabelian group. In other words, operations do not commute. Let x be a 90° rotation about the x axis and let z be a 90° rotation about the z axis, and note that xz is not the same as zx. The die is not in the same orientation.
We usually follow the faces when defining the group, but one could follow the corners instead. If you know where the corners are, you know where the faces are, and vice versa, so these two groups are isomorphic. The group is either embedded in S6, permuting 6 faces, or in S8, permuting 8 corners. To illustrate, let's check the size of the group again, this time using corners. Any of 8 corners can be placed in the top left front position. Once this is done, spin the cube about its diagonal and any of 3 corners can be placed in the top right front position. This fixes the cube, hence there are 8×3 possibilities.
As you might expect, the same group can be embedded in S12, as we focus on the edges of the cube. Place any of the 12 edges at the top left, and only 2 edges can appear at the top right. That's 12×2 = 24 orientations.
Sometimes we are interested in the group of reflections and rotations. Return to the faces of the cube. Any of 6 faces can be placed on top, then spin the die to move any of four faces to the front. Finally, we might reflect the die through a mirror, which swaps the left and right faces, and leaves the other four faces where they are. That's 6×4×2 = 48, twice as large as the earlier group.
Rotation and reflection groups exist for all the regular and semiregular solids. These groups can be described by following the faces, edges, or vertices.
You'll notice that the rotations of the octahedron are isomorphic to the rotations of the cube. Follow the faces of the octahedron and the corners of the cube simultaneously. These shapes are dual to each other, hence the faces of the octahedron correspond to, and move in lock-step with, the corners of the cube. You can even embed the cube inside the octahedron, so that the corners of the cube are the centroids of the faces of the octahedron; then move both shapes together. Similarly, the corners of the octahedron define the same group as the faces of the cube.
A truncated tetrahedron has the same rotation group as the tetrahedron. Place any of four faces on the table, then turn any of three faces forward. It doesn't really matter whether the sharp points are cut off or not. There are still 12 distinct orientations, or 24 if we allow reflections.
There are two shapes that cannot be reflected, the snub cuboctahedron and the snub icosidodecahedron. These exhibit chirality, as discussed earlier. Thus the reflection canot coincide with the original.