If by "The Western and Eastern Roman Empire" you mean the state of the Roman Empire in 5th century AD when it was usually ruled by two emperors, the answer is that it was theoretically a single state, ruled by two imperial colleagues: the Emperor of the East and the Emperor of the West. While we usually track this division from the death of Theodosius I in 395 AD, the fact is that the tradition of having co-rulers comes from the republican times when there were two consules elected for each year. In the imperial times, there was an early attempt to introduce two emperors by Tiberius who appointed two joint heirs: Caligula and Gemellus, but the former got the latter killed and remained the sole ruler. In the following century, there was a much more successful example of imperial colleagues: Marcus Aurelius and Lucius Verus. Arguably the first case in which we could distinguish the Emperor of the West and the Emperor of the East is the joint rule of Valerian (East) and his son Gallienus (West) although they did not include this geography in their titles at that time.

It was after the reforms of Diocletian that the Empire started to be ruled by multiple emperors who were in charge of a specific territory. Diocletian tried to introduce a model known as tetrarchy, where there were four imperial colleagues: two senior ones - augusti, and two junior ones - ceasars. The model did not last long, but the practice of having the Empire run by joint emperors did: sometimes there was a single emperor (like Constantine I) sometimes three of them (Constantine's sons), but most often there were two Emperors: one in the East and one in the West. The practice ended in 480 A.D. when the last emperor of the West Julius Nepos was killed and the Emperor of the East Zeno remained the sole ruler of (what was left of) the Empire.

In other words, the Empire was one in this period - it was just ruled by two Emperors. The constitutional laws were common: for instance, the code of Theodosius II was adopted by both the East and the West, although he was the Emperor of the East. The military was commanded from two centers, but in both cases it was Roman and in multiple cases the troops from both halves of the Empire fought jointly against common enemy (wars against the Vandals in Africa are the best example of this). To quote J. B. Bury:

"A few words may be said here about the unity of the Empire. From the reign of Diocletian to the last quarter of the fifth century, the Empire is repeatedly divided into two or more geographical sections — most frequently two, an Eastern and a Western — each governed by its own ruler. From A.D. 395 to A.D. 476, or rather 480, the division into two realms is practically continuous; each realm goes its own way, and the relations between them are sometimes even hostile. It has, naturally enough, proved an irresistible temptation to many modern writers to speak of them as if they were different Empires. To men of the fourth and fifth centuries such a mode of speech would have been unintelligible, and it is better to avoid it. To them there was and could be only one Roman Empire; and we should emphasise and not obscure this point of view."

The story about the two empires makes much more sense after the corronation of Charlemagne in 800 A.D. From that point, there are two distinct states with different laws, customs and civilization.