I am reasonably competent at using commutators to find my own algorithms on twisty puzzles. Many good tutorials for this exist online.
However, there are also algorithms that are not commutators, because they are odd permutations. These are usually used to solve some kind of parity problem or another. Examples include swapping two corners on a 2x2 Rubik's cube, the "J permutation" on the 3x3, the move that solves the last four edges on a 4x4, and the edge parity move on the 4x4.
I have asked several questions recently about specific examples of these, but each example still feels like a special case and I don't feel I have a deep understanding. For commutators the theory is easy to understand and with a bit of practice I can come up with my own easily, even on a puzzle I haven't seen before, odd permutations it seem much more of a black art and I haven't seen any tutorial or explanation of how to come up with algorithms for them. I am asking for any tips or theory that will allow me to construct my own.
I do know that in general one can often perform a single twist (which on a cube-shaped puzzle is a 4-cycle) and then solve the cube again using commutators, but, for me at least, this tends to result in a long and laborious sequence of moves, rather than a simple algorithm. So I am looking for any theory or techniques by which non-commutator algorithms can be constructed.