Even with Goat Head or Eucharist both rooms will not stay open, and entering either of the rooms will make the other disappear.
If Duality was bought through a Devil deal, it is still possible to get Angel Rooms as well as Devil Rooms.
Teleportation via a Joker card, a Red Chest or a Sacrifice Room before defeating the boss and choosing a door will follow the normal rules of Devil/Angel Room generation, instead of a 50/50 chance.
The room that was chosen this way will be the only one that opens once the boss is defeated.
Teleporting after choosing a door will always take you to the same room.
Duality is one of only four items to appear in both the Angel and Devil item pools, the others being 7 Seals, Lil Delirium, and Void.
The Duality image appears to be similar in concept to the yin and yang symbol of Ancient Chinese philosophy, as the latter represents the concept of dualism as well. However, they don't quite appear to be exact in terms of the diameter format.