Flag size : 15x23cm
According to the history Michael VIII Paleologue adopted this symbol after he had reconquered Constantinople from the Crusaders in 1261. It represented looking towards the East (Asia Minor, traditional power center of the Byzantine-government in exile after the IVth Crusade) and theWest (newly reconquered land in Europe.)
The double-headed eagle had in the two centuries of Paleologue rule become identified not just with the dynasty but with the Empire itself and, more generally, with institutions and cultural ideas outside the Byzantine Empire that still remained centered on Constantinople. Most obvious of these is the Greek Orthodox Church, centered in theory in Istanbul to this day, and so it is not surprising that the Church would use the flag.