Empress Irene in Byzantium nearly managed a marriage alliance with the Frankish emperor Charlemagne in the 780s. Otto III married a Byzantine princess in the 1000s-1100s which won the southern part of Italy and opened the possibility of mutual recognition of both the Byzantine and Holy Roman Empires as "Roman" in a nod to a time in the late imperial era when Rome was split into eastern and western halves to better manage such crises as food shortages and incursions.

Thing about it is, even though they were speaking Greek by the 7th century, they kept many of the old Roman institutions, and continued to think of themselves as Romans. Its the "Western View of Civilization" that chooses to ignore those facts in favor of seeing Byzantium as an "alien" monarchy. This is the same "Western" civilization that in 1204 sacked Constantinople just because a) they weren't Catholic, and b) they were the wealthiest state in Europe....then made half-hearted efforts to save them from the Ottoman Turks.

The West choosing to reduce the actual Roman Empire i.e: Byzantium to a mere Greek kingdom was the result of a conflict over which state could actually claim the mantle of Roman (Holy Roman or Byzantine). Holy Roman Emperors, despite being elevated from mere Frankish and Saxon kings by the popes, were only borrowing the name of Roman because the old Roman Empire was still fresh in the memories of many in the old western empire, particularly in the Frankish period. But because any actual Roman administration in the lands of Germania was confined to the extreme south and southwest of the region, they only had the influence of Rome to build on. Byzantium by comparison had all the legal, political, military and economic institutions from the former Roman Empire and an unbroken tradition of dynastic rule which had started with Caesar Augustus (Octavian) and continued even past the point in 476 when the last Western emperor, Romulus Augustulus, was overthrown by a barbarian general.