G parity ($G=\mathcal{C}{e}^{-i\pi J_2}$)

The product of charge conjugation and a half-turn in isospin is G parity. Both conjugations act on the isospin generators by $(J_1,J_2,J_3)\mapsto (-J_1,J_2,-J_3)$, so their product commutes with every generator. The Schur lemma therefore gives a common G-parity value throughout each irreducible isospin multiplet on which the operator acts internally. It can classify a charged multiplet even though individual charged states are not eigenstates of charge conjugation.

For integer isospin $I$ and neutral-state charge conjugation value $\eta$, the entire isospin multiplet has $G=\eta (-1{)}^I$. Both charge conjugation and the second-axis half-turn reverse the first and third generators, so their product commutes with the full isospin algebra. Evaluate it on the $M=0$ state and use ladder connectivity to obtain the value for every $M$.

Each pion has G parity minus one, so an $n$-pion state has $G=(-1{)}^n$. In the isospin-symmetric strong interaction, a parent with positive G parity can decay only to an even pion multiplicity, and a negative parent only to an odd multiplicity. The rule does not forbid small isospin-breaking or electromagnetic amplitudes.

The neutral pion has positive charge conjugation eigenvalue and its spin-one isospin half-turn has eigenvalue minus one. Hence the G parity eigenvalue is minus one. Since G parity commutes with isospin, both charged members have the same value. Charged-state phase changes alter individual charge-conjugation and rotation formulas together, leaving this eigenvalue unchanged.