Here is a visualization for n=8 of how many of these mutual assignments (which amount to edge 3-colorings of pairs of binary trees joined at the leaves, which is a little bit closer to cell 4-colorings of planar graphs):
Darker pixels mean more possible assignments. The 4-color theorem is the statement that throughout the triangle, it never gets completely white. Just think! If you could come up with a simple closed form description of this sort of diagram, you might have a nice short proof of 4CT.