Symmetry in physics has been generalised to mean invariance—that is, lack of change—under any kind of transformation.
So maybe the idea of symmetry is just an idea that is passed through on the learning journey—more about a change of approach than new substantive content.
A search for certainly can be as construed as a search for what does not change. The idea of a constraining relationship might provide a stepping stone.
It’s about spotting patterns of change: vary something, find something else does not change. Some patterns of change lead to insights, others don’t.
Aiming for an egalitarian treatment of all points of view: to treat identically is to treat symmetrically, which implies that there is a symmetry.