Ok. This coincidence feels like the key to helping me understand the equation in the title of this question. https://physics.stackexchange.com/questions/819552/how-to-prove-ei-theta-2-hatn-cdot-sigma-sigma-e-i-theta-2-hatn
Namely that we can implement a unitary transformation of a pauli vector either by conjugating by exponentiated pauli matrices, or by doing regular multiplication with a rotation matrix (the matrix exponential of a 3x3 skew symmetric matrix)