I love how some people call the HRE the legitimate Rome, while it has no ties to Rome. It's core territory, Germany, was never controlled by Rome. It was not Holy, as it had an tendency to clash with the pope. It was not Roman, but german. It was not an empire, but a decentralized kingdom.
The Byzantine Empire was the legitimate successor to the Romans. Just because some odd historian called it 'Byzantine empire' doesn't mean it's not the Eastern Roman Empire. It's inhabitants were mostly greek, but they regarded themselves as being Roman.
Moscow regarded itself as the Third Rome as they were Orthodox(which the second Rome, Constantinople, also was). They also had a marriage to a Byzantine princess.
The HRE was recognised as the Western Empire by the Byzantines several times, and the very term "Emperor" implied that it was Roman back in medieval times. The HRE was an empire for a time, and even if its grip on power faded, well, so did Byzantium's.