Independent head-tail reversal symmetry (${\mathbf{s}}_i\mapsto {\sigma }_i{\mathbf{s}}_i,{\sigma }_i\in \{-1,1\}$)

An apolar axis model with pair energy $-({\mathbf{s}}_i\cdot {\mathbf{s}}_j{)}^2$ is invariant when each molecular representative is independently reversed. The vector average changes under these replacements, but the nematic order parameter made from second moments does not. In an unbiased statistical ensemble the vector average vanishes; an individual finite representative need not have zero average. For independently chosen signs of a perfectly aligned axis, $⟨|N^{-1}{\sum }_i{\mathbf{s}}_i{|}^2⟩=1/N$. This distinguishes exact head-tail invariance of an observable from vanishing only after an average.^[ref]