Half-turn phase of an SU(2) multiplet (${\alpha }_M=(-1{)}^{I-M}$)

With positive ladder coefficients, the second-axis half-turn is $e^{-i\pi I_2}|I,M⟩=(-1{)}^{I-M}|I,-M⟩$. Conjugation sends $I_{\pm }$ to $-I_{\mp }$, making consecutive phases differ by a minus sign. The highest-state rotation formula, expanded through its last nonzero lowering term, fixes the highest-state phase to one.