I would like to extend the method to verify isomorphism between cubes with colored faces in this answer to $4$-cubes (tesseracts) with colored faces ($2$-faces), allowing rotations and reflections, and if possible, extend it even further to $n$-cubes with colored $2$-faces.
However I am unsure if in the case of the tesseract, taking all the couples of colors of the opposite faces for all the $8$ composing cells ($8$ cubes) is sufficient to identify uniquely the tesseract. Maybe we need also to consider "opposite" cubes in the tesseract, but I am unable to figure it out. Any hint?