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After playing for a while, I figured out that you have a choice to trigger the checkpoint, or jump it (previously I was jumping it every time). The game tells you that you can skip it "for a greater challenge", and I see that I get some points at the end of the level for doing so.
I've already found that there is a Perfect and Perfection+ outcome for each level, and I've read about "Double Perfections" and "Triple Perfections" in other questions - so I'm curious if there is any other "collectible/challenge" reason to skip the checkpoints. By the time I've done a few passes through the game I don't want to have to do it all again if there is some special outcome for skipping them.