Does it hold that $F \equiv \sigma(F)$ for a CNF formula $F$ and a permutation $\sigma$ s.t. $F \vDash \sigma(F)$?
The crucial observation is that if $A \vDash B$ then also $\sigma(A) \vDash \sigma(B)$. This follows since all $\sigma$ does is rename variables and flip some variables. For example, if $\sigma(x) = \lnot y$, $\sigma(y) = z$, and $\sigma(z) = x$, then $A(x,y,z) \vDash B(x,y,z)$ implies also $A(\lnot y, z, x) \vDash B(\lnot y, z, x)$.

Let $G$ be the set of all permutations of literals satisfying your condition. I claim that $G$ forms a group with respect to composition. Since $G$ is finite and composition is associative, it suffices to check that it is closed under composition. Indeed, if $\sigma(\lnot x) = \lnot \sigma(x)$ and $\tau(\lnot x) = \lnot \tau(x)$ then $\sigma(\tau(\lnot x)) = \sigma(\lnot\tau(x)) = \lnot\sigma(\tau(x))$.

Since $G$ is a group, $\sigma^{|G|}$ is the identity. If $F \vDash \sigma(F)$ then, applying $\sigma$ on both sides, we get $\sigma(F) \vDash \sigma^2(F)$. More generally, we get $\sigma^n(F) \vDash \sigma^{n+1}(F)$, and so $\sigma^n(F) \vDash \sigma^m(F)$ whenever $m \geq n$. In particular, choosing $n = 1$ and $m = |G|$, we obtain $\sigma(F) \vDash F$.

The group $G$ is known as the signed symmetric group, denoted $B_n$, where $n$ is the number of variables.