Tipsy Trip

The faces of a cube are adorned with squares, circles, and crosses. As seen in the mirrors, opposite faces show the same shape (and color), but one is filled and the other is hollow.

The cube sits on a 6x6 grid whose spaces have the same symbols, as shown. It moves about the grid, space by space, by tipping over an edge so that a face lands on a neighboring space with the same symbol. (Tipping over a corner is not allowed; nor is “spinning in place”.) 

Find the path through the grid that uses the smallest number of steps from start to end.

You can find a printable version including a cut-out pattern to make a physical cube here.