I've found multiple very interesting answers regarding the cultural and identity differences between the inhabitants of both states. But in this case, I'm more interested in how did they approach it from the diplomatic/legal point of view. I.e both empires signed multiple alliances and treaties with each other (and I assume there was significant diplomatic communication in general between them) over the centuries.
Was their relationship at any point similar to the one between Eastern and Western Roman empires in late antiquity (both recognized that at least formally they constitute a single undivided state which is ruled by two emperors). Or did they always saw one other as independent states, and just ignored the fact that legally they considered each other to be illegitimate due to practical reasons?
In the second case, did their emperors always refer to each other as the "Emperor" of the Greeks or Germans/Franks/Latins?
At first, the Byzantines probably thought the HRE was kind of a restoration of the western Empire. If so, in their mind, the emperor in Constantinople was of course the "senior" emperor. And they did manage to get along sometimes, especially when a Byzantine princess, Theophanu, married the Holy Roman Emperor Otto II.
But eventually both sides had to deal with the fact that the western emperor considered the eastern emperor illegitimate. Charlemagne had been crowned in 800 because they believed the eastern throne was vacant - partly because it was held at the time by a woman, Irene, but also because Irene had overthrown her own son.
By the 12th century, when the Holy Roman Emperor tried to pass through Constantinople during the Third Crusade, they definitely did not recognize each other's claims.
I have some previous answers that may be helpful...if they don't answer your specific questions, just let me know and hopefully we can help clarify things further
What was the Eastern Roman reaction to Charlemagne being crowned Roman Emperor by the pope in 800?