3-SAT where variables occur equally many times as a positive literal and as a negative literal
It is NP-complete (but I don't know if it has a name): suppose that a variable $x_i$ appears as a positive literal $n$ more times than as a negative literal.

Then you can "balance" it adding $n$ new 3CNF clauses with $n$ new variables $y_1,...,y_n$:

$-x_i \lor y_1 \lor -y_2$
$-x_i \lor y_2 \lor -y_3$
...
$-x_i \lor y_{n-1} \lor -y_n$
$-x_i \lor y_n \lor -y_1$

If $x_i$ appears more times as a negative literal, apply the same expansion but using $x_i$ in the new 3CNF clauses instead of $-x_i$.

The $y_i$ are balanced and the resulting formula (that can be built in polynomial time) is clearly satisfiable if and only if the original 3CNF formula is satisfiable: whatever is the value of $x_i$ the new clauses can be satisfied setting $y_i = true$, so they don't "interfere" with the satisfiability of the original formula.