I just realized that if you hold down z from the moment you die, you can play the exact same level in terms of dye & disc placement. Here's a demo where I play the same Kiwi layout several times in a row:

Edit: Actually, it seems that you don't get to repeat the same level. Rather, having the instantaneous restart always gives you the same level shown here. My guess is now that the levels are seeded based on how long the title screen is shown before you select a level.

Edit 2: On second look, it seems that the initial seed changes when you start the game fresh or reset it, using either the pause menu or shift+f. And it seems that the seeding starts from the moment the "Your Score" screen starts to load.

Edit 3: Which means it is useful for Orange after all!