Does “$\forall x\in L, \sigma(\neg x)=\neg \sigma(x)$” hold given that $\sigma(F)\equiv F$ for a CNF formula $F$ built on a set $L$ of literals?

Suppose we have a CNF formula $ F$ built on the set of literals $ L=\{x_1,\neg x_1,\cdots,x_n,\neg x_n\}$ where each variable is used in at least one clause of $ F$ . Consider a permutation $ \sigma$ of $ L$ such that $ \sigma(F)$ is logically equivalent to $ F$ i.e. $ \sigma(F)\equiv F$ .

Does it hold that $ \forall x\in L, \sigma(\neg x)=\neg \sigma(x)$ ?

I tried to find a counter example without success.