Searching for name of equivalence property in hamiltonian paths
The 2-OPT heuristic for finding TSP tours is fairly close to what you're asking about. It's a local improvement algorithm which looks for sections of a TSP tour which, when reversed, will yield a shorter tour. The 3-OPT and the Lin-Kernighan heuristics work along the same lines, but use more complicated moves. 

Looking for the optimal rearrangement of some subpath of a TSP tour is going to be inefficient unless it only involves a short section of this tour, and then I imagine it will be inferior to more global rearrangement algorithms such as 2-OPT (which can reverse a large subpath of a tour).

As far as I know, the concept that you're asking about has no name.