I have beaten Isaac the Cathedral level once, and when I returned to the title screen, there was a rectangle with what appeared to be a piece of a puzzle in it. I infer that I can complete the puzzle by beating Isaac with each character. Is this the way to complete the puzzle, and what happens when I complete it?
Your guesses are correct. The puzzle is completed by beating the Cathedral with each character.
Here it is after my second victory against Isaac:
Completing the puzzle unlocks the Polaroid trinket, an item that shields Isaac when brought to 1/2 heart, but that also