Conversion between k-SAT and XOR-SAT
If all XOR relationships between variables in CNF formulas could be detected in polynomial time, then this would allow the solution of UNAMBIGUOUS-SAT in polynomial time.  By the Valiant–Vazirani theorem this result would imply that NP = RP.

To solve UNAMBIGUOUS-SAT, recall that $a \oplus b$ implies $a \neq b$.  Find the XOR relationship between each pair of variables and use the results to divide the variables into two groups of equivalent variables.  Once this is done, only two test assignments are required to determine satisfiability.

In the limited case of recovering XOR relationships encoded in the usual way, i.e.

$a \oplus b \oplus c$

to

$
\lnot a \lor b \lor c \\
a \lor \lnot b \lor c \\
a \lor b \lor \lnot c \\
\lnot a \lor \lnot b \lor \lnot c
$

this can be done in polynomial time by sorting the clauses followed by a linear-time scan.