= Exchange parity of two coupled identical spins
{title2=$P_{12}|(ss)jm\rangle=(-1)^{2s-j}|(ss)jm\rangle$}
Exchanging the factors in the <Clebsch-Gordan decomposition> of two equal <spins> gives the phase $(-1)^{2s-j}$. This follows from the exchange symmetry of the <Clebsch-Gordan coefficients>. The total exchange parity includes the spatial state and any other internal factors: the spin parity alone does not impose the full <symmetrization postulate>. For two half-integer <spins>, even $j$ is spin-antisymmetric and odd $j$ spin-symmetric.
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