The standard problem 1-in-3 SAT (or XSAT or X3SAT) is:
Instance: a CNF formula with every clause containing exactly 3 literals
Question: is there a satisfying assignment setting precisely 1 literal per clause true?
The problem is NP-complete and remains hard even if no variable occurs negated. I wonder whether this problem becomes easy or remains hard if each variable is required to occur at least once positively and at least once negatively.
The usual reduction from 3SAT showing that 1-in-3 SAT is hard replaces a clause $(x\lor y \lor z)$ by clauses $(\lnot x \lor a \lor b)$, $(y\lor b\lor c)$, $(\lnot z \lor c \lor d)$ where $a,b,c,d$ are fresh for each clause. Thus, this reduction doesn't help in answering my question. I've had trouble coming up with a gadget showing hardness of this variant, since if exactly 1 literal in a clause is true, then non-symmetrically 2 literals are false. If it turns out to be easy, thinking in terms of partitions of the clause set might do it, but I can't see how.