I have two right-handed coordinate systems that are off by some unknown transformation. I know this transformation is "simple": It consists of only rotations of 0/90/180/270 degrees around the standard $x$-, $y$-, and/or $z$-axis. I want to enumerate all such "simple transformations" possible.
If there is just one rotation around one axis, then I know I have at least $4\times3=12$ such transformations, where $4$ is the number of possible rotation angles, and $3$ is the number of possible axes ($x$-, $y$-, and $z$-axes).
But apparently, we may have 2, 3, 4, ... rotations around 2, 3, 4, ... axes, up to a point where we start to have repeated transformations that can be reduced to a smaller set of rotations that produce the equivalent rotation.
How many unique "simple transformations" do we have? How do I enumerate them all?