Inspired by A closed path on the Rubik's cube.
Let $N$ be a positive integer. Consider an $N\times N\times N$ cube, with each face tiled by $N^2$ squares measuring $1\times 1$. A closed path is drawn on the surface of the cube, never entering the same square twice, never passing through the corner of a square.
Each time the path enters a square, necessarily through an edge, it will exit through a different edge. Let $R$ be the number of squares with exit edge immediately to the right of the entry edge, and let $L$ be the number of squares with exit edge immediately to the left of the entry edge. Ignore squares not meeting the path, and squares for which the exit edge is across from the entry edge.
What are the possible values for $R-L$?