What a strange use of the word "symmetric": "Two grid paths are symmetric if they share the same start and end point and one can be derived from the other by swapping the order of the constituent vectors."
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.