The double-headed eagle

The double-headed eagle was the symbol of the Paliologoi, the last Greek-speaking "Roman" (i.e. Byzantine) dynasty to rule from Constantinople. The Emperor Michael VIII Paliologos. recaptured Constantinople from the Crusaders in 1261, from a state based in Asia Minor; the double-headed eagle symbolized the dyansty's interests in both Asia and Europe, and was kept despite the fact that virtually all of the Asian possessions were gobbled up by the Ottomans within a generation of the recapture of the City. Michael's descendants stayed on the Byzantine throne until the City and the Empire fell to the Ottomans in 1453.

The double-headed eagle had in the two centuries of Paliologan rule become identified not just w/the dynasty but w/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.