How many distinct rotations are possible in ND? This is the same thing as asking how many distinct magnets are possible.

There is a crucial distinction. Is it possible to identify the different rotations? It makes a difference. This is true even here in 3D. If you have no way to orient yourself then there is only one possible rotation in 3D. If you look at the Earth from above the north pole then the earth is rotating counterclockwise. If you look at the Earth from above the south pole then the earth is rotating clockwise. If on the other hand there is some way for the observer to orient themselves, by using the magnetic field or stars or whatever, then there are two possible rotations.

In 2D there are two distinct rotations, as the 2D observer can't help but have an orientation. So let's exclude that and consider the non-orientable observer cases. In 4D there are also two, as the two rotations can be compared against one another. In 5D only one. Essentially the distinction is that in odd dimensional spaces there is always an axis, a null dimension that isn't rotating. One can use this dimension to rotate so that any given rotational plane appears to change direction.

So how many distinct rotations are there in a 2N dimensional space without orientation? There are always two. The only invariant is the parity of the rotations. Any rotation of the observer may change the sign of two rotations, leaving parity the same.

Now suppose it is possible to order the planes of rotation. Perhaps the rotational periods give us an ordering. Then there are 2^(N-1) distinct rotations for 2N > 2. Same with the possible number of types of magnets. If the magnetic planes are of differing strength then there are the same number of possible different magnets.

If you have some way to orient the first plane in the order, then you get a full 2^N distinct rotations.