@nprofile1q... - When I said "Surprisingly, there are three basic ways to 'rotate' a hydrogen atom" I was talking about the 3 commuting SU(2) actions whose generators are (L + M)/2, (L - M)/2 and S, where L is the orbital angular momentum, M is a version of the Runge-Lenz vector, and S is the spin angular momentum. Of course this isn't "surprising" to physicists - I'm sure Pauli knew it, since he knew about L and M in 1926 and his paper back then hinted at S, which he later discovered. But it seems painfully under-discussed in physics textbooks. And lot of mathematical work on the Kepler problem, which rejoices in the existence of M, ignores spin.