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."
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.
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.
