General structure of solutions to 3-SAT circuits
The theory you are after is universal algebra. See the excellent expository article of Hubie Chen, A rendezvous of logic, complexity, and algebra, which contains a streamlined proof of Schaefer’s dichotomy theorem, which was recently extended to arbitrary alphabets.

3SAT has no polymorphisms. NAE-3SAT has only negation as a polymorphism. All other classes of polymorphisms correspond to CSPs solvable in polynomial time. See Allender et al., The Complexity of Satisfiability Problems: Refining Schaefer’s Theorem.