The modern double-headed eagle flag for the Ecumenical Patriarchate of Constantinople and of Mount Athos, features the eagle with a cross in the right claw symbolizing spiritual authority, and an orb in the left symbolizing secular authority.
Above the eagle, is a crown, and the background colour of the flag is yellow or gold. This flag is often used also by the Greek Orthodox Churches in the diaspora under the Patriarchate as their official flag (such as in America, Canada, Australia, etc), and is not to be confused with the double-headed eagle used by the Church of Greece. Having said that, many monasteries and churches in Greece do currently fly this golden/yellow flag, for historical reasons.
The yellow with a black crowned double-headed eagle flag, was the symbol of the Paleologues, the last Greek-speaking “Roman” (i.e. Byzantine) dynasty to rule from Constantinople. Emperor Michael VIII Paliologos recaptured Constantinople from the Crusaders in 1261, from a state based in Asia Minor; the double-headed eagle symbolized the dynasty’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.
This flag had in the two centuries of Paleologan 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.
Emperor Isaakios Comnenos (11th century AD), the first ruling member of the Comnenus dynasty, was the first Emperor who adopted the two headed eagle as the symbol of the Empire which comes from the single headed imperial eagle motif, traditional for the Roman Empire.
— Source: asceticexperience.com