I'm not sure if I should analyze the modal verb need with the particle to or without it in this sentence: We need to clean our room.
With the modal verbs must and can, there is no to: We must clean, We can clean. So I'd say the to belongs to the need.
Another way to analyze this is to make hypothetical sentences like these:
Clean the room if you must.
Clean the room if you can.
Clean the room if you need to.