I can see how defining symmetry for a pathfinding domain might be non-intuitive. What the quoted definition is trying to communicate is that paths sharing such a relation are permutations of one another and thus, symmetric.
Seeing the permutation property isn't straightforward until you change your definition of a path: from an ordered sequence of edges to an ordered sequence of vectors.
Take the following two paths as examples:
p1 = {up, up, up, right, right, right} p2 = {up, right, up, right, up, right}
Not only are they equivalent but I can derive one from the other by just changing the order of the moves.
Such symmetries are plentiful on grid maps: as soon as you have a large open area, you introduce lots of possible ways to cross it.