The article gives a concise history, from a Greek Orthodox perspective (with emphasis on both Greek and Orthodox), of how the double headed eagle came to have the meaning it has today. Below is my translation of the Archimandrite’s words (apologies for the run on sentences but, that’s how the original is too):
It has always been that people used places, shapes, images, plants and animals as symbols to express an idea. These symbols, called hieroglyphics, are found in all populations and all religions. In those, the imagination of the people borrowed much from the kingdom of the plants such as the olive, the laurel, the lilly, the pomegranate, etc. These were used to express different meanings and ideas.
Also, since antiquity all peoples used as an expression of power, force and majesty the Lion. As an expression of height, poetic elation, wisdom and ingenuity was the Eagle. For this reason the Lion is used in the palaces of kings and the heraldry of nations and sovereign Houses, whereas the Eagle is found on the friezes of temples. Lions decorate the palaces of the kings of Babylon and Persia. Lions decorate as well the palace at Mycenae. Lions [in sculpture] were raised in Marathon, Thermopylae, Chaeronea, etc.
The Eagle, on the other hand, is the symbol and the messenger of the Olympian Zeus. An Eagle transfers Ganymede to Olympus. An Eagle eats, on Zeus’ orders, the liver of Premetheus who is bound to the Caucasus. An Eagle is places on the frieze of the temple of the Olympian Zeus and on the temple of the Capitol Zeus. With the meaning of ingenuity and inspiration, the Eagle was given as a symbol to the Apostole and Evangelist John the Theologian.
Ptolemy Lagus of Alexandria first placed the Eagle on the flags of his army as a symbol of the state and military power. From the Ptolemaic state of Alexandria, dissolved by the Romans, did the Roman Emperors take the Eagle for their military flags. Slowly, the Roman Eagle became the symbol of the entire Roman Empire. As the Roman Empire was split during the reign of Diocletian and Maximilian into two states, the Eastern and the Western, so was the Eagle split into two. The Eagle of the Western Empire had its head facing west and the Eagle of the Eastern Empire faced East. When the Roman State, through Constantine the Great, was united into a single Empire, the two symbolic Eagles were reunited into one. However, the previously single headed Eagle became double headed, symbolizing the unified empire with the two capitals: Old Rome and New Rome. For this reason one of the heads faced West towards Old Rome and the other faced East towards New Rome – Constantinople.
When the Emperor Constantine the Great saw in the sky, at high noon, the divine symbol of the cross with the epigraph ΕΝ ΤΟΥΤΩ ΝΙΚΑ [translated to Latin as "IN HOC SIGNO VINCES"], he was instructed in a vision where he was visted by Jesus Christ to construct a copy of the image of the cross and to use it throughout his military ranks so that he may be victorious through it. This is when Constantine the Great, according to the instructions he received, constructed the Labarum and replaced the until then national military flag of the Double-headed Eagle with the new Christian banner of the Cross. However, he maintained and established the Double-headed Eagle as the symbol and state emblem of the Imperial Dynasty of the Byzantine Roman State. In the following centuries of the height of the Byzantine State, the Double-headed Eagle was used as the Coat of Arms of the Byzantine Emperors, placed on the Imperial Palaces, decorating the chests of the imperial guards and the courtiers in general. It was embroidered on the imperial clothes, the saddles of the imperial horses and on the imperial shoes. In the sacristy of the Holy Monastery of Great Lavra on Mount Athos, is preserved the imperial coat with Double-headed Eagles embroidered in gold, once belonging to the Emperors Nicephorus Phocas and John Tzimiskes from the 9th century.
After the Fall of Constantinople to the Turks (1453) the Ecumenical Patriarch of Constantinople was recognized by the conqueror Muhammed II as the Ethnarch [national leader] of the Romans. As such, he inherited along with the other imperial symbols (the crown, sceptre, robes) the Double-headed Eagle as the symbol of the Nation. This symbol has been used ever since as the emblem of the Ecumenical National Patriarchate and used in the seals of the Patriarchal Bulls. Because of this, it is carved as the Coat of Arms above the gate of the Patriarchal churches. It is also carved on the walls and the floors of the Patriarchal churches and stavropegial monasteries and those houses made stavropegial. It is also conserved on the floors of many temples among which is the temple of the Metropolis of Mistras. It is embroidered in the Patriarchal clothing and robes. Embroidered on fabrics or tapestries, wherever Constantinople is depicted. In time, whenever a cleric in the jurisdiction of the Ecumenical Patriarchate is ordained and the floor of the particular temple does not have the Double-headed Eagle carved, a tapestry with the symbol is placed under the feet of the ordained to show that he is stepping and is on Patriarchal ground, as spiritual subject of the Patriarchate.
As a Patriarchal emblem, the Double-headed Eagle became part of the Patriarch’s vestments. It is curious how this particular item did not become more widely adopted by other Bishops like the other imperial emblems did such as the Crown (mitre), the Purpure robes and the Sceptre (pastoral staff).
The spirit of the Greek people maintained alive throughout the years after the Fall [of Constantinople] the Byzantine tradition of the Double-headed Eagle and in the expression of Greek art. The Double-headed Eagle, that before the Fall has such symbolic value, did not lose it later. On the contrary, it was conciously kept as the symbol of the continuing Byzantine tradition and became the premier symbol of the national ideal of the recreation of the Great Empire.
For this reason, it is the most beloved theme of neohellenic art and has an excellent place there. How closely tied is the Double-headed Eagle with the popular concious [of the Greek people] is shown the innumerable depictions in all forms of Greek popular art whether it is in textile, woodwork, metalwork, jewelry etc. Even in our own century [20th] the wave of nationalism raised the Double-headed Eagle to the first and most honored position. It is still carved onto many objects of the Church and the residential furniture, woods, silver and gold objects. It is painted on many icons and carved onto seals. The seal of the Archbishopric of Athens as well as many other Metropolia use the Double-headed Eagle. It is also depicted on every other form of art upon which a national meaning is desired to be given.
–Archmandrite Zacharias A. Lianas