To quote u/Tiako

They saw each other as one nation that happened to have two administrative centers. I would argue that they never fully split--until the West went into freefall around 455 they were ruled by the same family, and freely meddled in one another's affairs. This included the loan of troops by the East to reconquer Africa after it fell to the Vandals.

The Western and Eastern halves of the Roman Empire were not, as you suggest, "two empires," but divisions of the same empire. My home state is made of townships, which compose counties, which compose the state, which then composes the country. Likewise, Rome's largest division was once provinces. The Emperor Diocletian, at the end of the 3rd century, realized that the empire was too large to rule as one emperor, so he grouped the provinces into four larger divisions, which later resolved into just two.