I’m interested to hear about anime that give a different experience when putting the episodes in a different order. They can be official orders (broadcast vs dvd) or fan orders (moving arcs around, putting seasons in chronological vs release order). Maybe you can even think of examples where a companion movie is inserted in the middle of a season? Two separate projects by the same author or studio that can be weaved together to form a consistent whole?
Hidamari Sketch. The first two seasons have an achronological order. Each episode is a day in their life in one of the twelve months. Online you can find alternative orders that if you follow them, the story is told chronologically. Although I don’t think it would make that much of a different experience given the episodic nature and lack of overall narrative. One thing about the release order is that sometimes the characters refer to something that has happened but the viewer hasn’t seen that yet. That is something that would disappear with the alternative order. From what I heard the manga is mostly chronological so it would be more true to the manga.