= Independent head-tail reversal symmetry
{title2=$\mathbf s_i\mapsto\sigma_i\mathbf s_i,\quad\sigma_i\in\{-1,1\}$}
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, $\langle|N^{-1}\sum_i\mathbf s_i|^2\rangle=1/N$. This distinguishes exact head-tail invariance of an observable from vanishing only after an average.<Lectures on Landau Theory of Phase Transitions by Peter D. Olmsted>[§4.3, printed pp.10–11, molecular head-tail equivalence]{ref}
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