Are rotations considered the same as object spatial orientations, mathematically speaking?
Say, doing 20 full rotations around an axis is the same as doing 2 rotations or 50 rotations, it leaves you at the same orientation, but these are still different operations.
Similar with doing a 180 turn clockwise or counterclockwise, you end up at the same position but the operation is different.
Seems to me like a rotation is not just the end motion but also defined by the path taken. Any mathematical object tells apart general rotations from orientations?