 Pauli-string commutation parity

For tensor products of Pauli factors from $\{I,X,Y,Z\}$, let $k$ count the sites where both factors are nonidentity and different. Distinct nonidentity Pauli matrices anticommute, and equal or identity factors commute. Reversing the product site by site therefore gives $PQ=(-1{)}^{k}QP$. Thus two Pauli strings commute exactly when $k$ is even. The criterion applies to the perfect-correlation observables in the GHZ theorem.