Mathematically, a necessary condition for any matrix to be a proper rotation matrix is det(U) = 1. Now, if we ever find that, in practice, det(U) ̸ = 1, we may conclude that there must be one (or an odd number of) negative eigenvalue(s) corresponding to an eigenstate that flips its sign from the previous geometry, meaning that the phase of the adiabatic state is not continuous.