The grid below can be partitioned into several regions, each of which is symmetric in some way (be it rotation or reflection). A cell with a number indicates the length of the shortest walk, within that region, to an “opposite” cell in the region. (I.e., a cell that could potentially be moved to the location of the numbered cell under a non-trivial rotation or reflection of the plane.)
(A few examples: In a 1-by-N region, all cells could be labeled 0, since a reflection across the longer mid-line takes every cell to itself. In the “S”-shaped Tetris piece, the cells could be labeled 3, 1, 1, 3.)
The answer to this puzzle is the sum of the cubes of the areas of the regions in the completed grid.