In fact, when Karl the Great was crowned roman emperor in 800 and later the German Kings since Otto I. (933) were crowned automatically shortly after becoming German King, the Eastern Roman Emperor feared, that these "new" ones would claim their throne as well, as Karl wanted a kind of restoration of the (Western) Empire. But nevertheless the fear was without reason, the relationships were quite good.
Additionally before 476, the fall of the Western Empire, there were in fact two Emperors since abou 392, one in Rome, the other one in Constantinople.
PaKo