"Insensitive" CNF/DNF SAT always satisfying same number of clauses
Consider a DNF $C_1 \lor \cdots \lor C_m$.

For any clause $C_i$, we can compute the probability that a random assignment satisfies $C_i$: it is 0 if $C_i$ contains both a variable and its negation, and otherwise it is $2^{-|C_i|}$, where $|C_i|$ is the number of literals in $C_i$.

For any two clauses $C_i,C_j$, we can compute the probability that a random assignment satisfies both $C_i$ and $C_j$: it is just the probability that a random assignment satisfies $C_i \land C_j$.

Given this data, it is routine to compute the variance of the number of satisfied clauses under a random assignment. The DNF is insensitive iff the variance is zero. The corresponding algorithm runs in polynomial time, and so the problem is in $\mathsf{P}$.

If we have a CNF, we can negate it to obtain a DNF which is insensitive iff the original CNF is.