Launch Base Zone 1's background uses a simple alphanumeric order, starting with the numbers 0 through 9, and then continuing from A, going up alphabetically until no more tiles are left.
When more than 36 tiles are necessary, most levels will attempt to squeeze two digits into a single tile. The pendulums in Hydrocity Zone only require six tiles, so the artist got cute and filled them with dominoes.
Probably most interesting are the stages that opt to fill in the placeholder tiles with katakana, one of the three Japanese writing systems. The pattern found on the Hydrocity Zone 2 waterfalls is simply a truncated gojūon, which much like the alphabetical order used above, consists of the Japanese characters in their natural order.
However, the characters present in the slot machine Bonus Stage background seem to have been chosen deliberately, though I can't figure anything out from them. Maybe someone more well-versed in Japanese could give it a try?