Two equivalent paths share the same endpoints and have the same length. Two symmetric paths also have these properties but they can also be shown to be permutations of each other.

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.