I have seen this thing under
the name "space shuttle" - in a cylinder like rubiks cubes used to
come in. One way to determine the color layou is to make
an assumption (as I did) and proceed to solve-- if you lose, you
get the "impossible" single-pair-swap. Then swap two edges
of your assumption and try again. Incidentally, the plural
of crux is cruces, not cruxi.