Some reading tonight suggested that the Byzantine empire continued to call themselves "Romans" or "Imperium Romanum" even after the center of power in Rome itself was destroyed/moved. Why is this? Did they continue to imagine the height of power could be restored after a reconquering of the west? Or was it as simple as the cultural practices had been so deeply ingrained that Roman was not related the city and heart of the former empire at all?
Edit: (Sorry for using the term Imperium Romanum, I know they spoke Greek instead of Latin, but I believe there were equal terms used in Greek)
Rome hadn't been the centre of power in the Roman Empire for a very long time by the time "Rome" "fell" in "476" (I use quote marks here because all three things are questionable). When Constantine made Constantinople his capital, emperors had already been basing themselves at Nicomedia, Trier, Milan, Sirmium etc. The city of Rome simply wasn't that big a deal any more, in terms of political power at least. Roman identity was also a much more inclusive one in late antiquity (and probably from much earlier as well), one that did not necessarily focus on the city itself; after all, people from both Britannia and Egypt can claim to be Romans, yet there was still a world of difference between them!
Nor did Rome really fall in 476. Western provinces had been falling beyond imperial control for decades by 476 and the eventual coup that finally brought down the regime nominally headed by the emperor Romulus Augustulus did not end the empire - the general Odoacer merely deposed one emperor and acknowledged the authority of Julius Nepos, a previously deposed western emperor hanging around in Dalmatia. When Nepos died, Odoacer transferred his nominal loyalty to Constantinople. The Roman Empire was never destroyed - instead, political authority in the western empire was in theory transferred first to Nepos and then to the eastern emperor Zeno. Zeno moreover was an emperor who can claim to be the heir of Constantine. Succession in the east was generally peaceful until the end of the sixth century and marriages with previous imperial dynasties further facilitated this. Zeno was not directly related to Constantine, but that hardly mattered, since very few emperors can claim direct descent from an earlier emperor throughout the history of the empire. Instead, political legitimacy, as provided by the people, the senate and the army, was more than enough to create a sense of continuity. This continued until 1204, when Constantinople was taken by Latin forces taking part in the Fourth Crusade. There was no need for the Romans in Constantinople to delude themselves - they were the Romans and all their neighbours recognised that fact. This image was only cemented further by empire's continuous involvement in western affairs, as New Rome's power and prestige was acknowledged by everyone in late antiquity, even by the the so-called barbarian kingdoms of the post-Roman West.
Rome the city had also been retaken by the emperor Justinian I (527-565), who had conquered it from the highly Romanised Ostrogoths. Indeed, it is very fashionable to blame Justinian for ruining everything right now. Capable rulers of Italy after 476, particularly Theodoric the Great, stabilised the region and adopted aspects of imperial authority, to the extent that the 'barbarian' Theodoric may have been seen by his subjects as a new emperor ruling a renewed western empire. Even if we disagree with this, there was only an interregnum of ~60 years between 476 and the restoration of a "proper" Roman government in Italy, since Justinian's reconquest once again placed Rome under imperial control, which would last into the eighth century. Either way, the Roman Empire did not die in 476 and plenty of people were right to consider themselves Romans.
The Eastern Roman Empire was just as much the Roman Empire as the Western one. Indeed, by that time, the Western Roman capital wasn't even Rome anymore, it was in Mediolanium. In fact, Justinian, one of the greatest Eastern Roman Emperors, made a concerted and huge effort to reconquer the whole empire, and was fairly successful even though those conquests did not last. So there isn't really a way to separate the Western and Eastern Empires in terms of how "Roman" they were.
I was recently reading Usamah Ibn-Munqidh's First Crusade era work "The Book of Contemplation" (a great read by the way if you are interested in the era), and although he refers to the Crusaders as the Franks, whenever he talks about the Byzantine empire, he calls them "Romans," and their emperor "the Roman emperor" which is a testament to how widespread and long-lasting the idea of the Byzantine Empire as the Roman Empire was.