Something interesting from Medievalists.net:

In Byzantine heraldry and vexillology, the double-headed eagle (or double-eagle) is a charge associated with the concept of Empire – the heads represent the dual sovereignty of the emperor both in secular and religious matters and/or dominance over both East and West. After the Holy Cross, perhaps no other symbol has been associated more closely with the history and fate of the Byzantine Empire than the double-headed eagle motif, to the point that it has been ‘chiseled’ in modern imagination as being the ‘official flag’ of the empire up to its dying days in 1453. However, how accurate is this association, and how informative our sources are about this?

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The single- and double-headed eagles both appear [in the Byzantine Empire] from around the middle 12th century onward in the decoration of buildings built by members of the imperial family of the Komnenoi, such as the single-headed eagle from the Theotokos Kosmosoteira at Pherrai, western Thrace, commissioned by the sebastokrator Isaakios Komnenos in 1152. The double-headed eagle appears commonly throughout the Palaiologan period, as for example in a well-known plaque from the Metropolis of Mystras in the southeastern Peloponnese.

However, this motif was not used exclusively in Byzantium, and we can see the two-headed eagle appearing in mosques, fortresses, palaces, and Anatolian Seljuk caravanserais as a magical (animistic) and protective symbol of strength. Mainly we see it in profusion during the reigns of the Grand Seljuk Sultans of Rûm Alaeddin Keykubad I (1219-1237), and his son and successor Gıyaseddin Kay Khusraw II (1237-1246). This usage declined sharply after the Battle of Köse Dağ in 1243, as many Seljuq traditions of pre-Islamic origin were abandoned, including the depiction of animals.

The Palaiologan emperors used the double-headed eagle as a symbol of the senior members of the imperial family. The emperor is always distinguished by his richly jeweled regalia, like in the famous Athonite chrysobull of 1374 where Alexius III of Trebizond wears purple and jewels, while his consort’s garment is decorated with double-headed eagles.

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Other Balkan states followed the ‘Byzantine model’: chiefly the Serbians, but also the Bulgarians and Albania under George Kastrioti (better known as Skanderbeg), while after 1472 the eagle was adopted by Muscovy and then Russia. The Ecumenical Patriarchate in Constantinople and Mount Athos, and the Greek Orthodox Churches in the diaspora under the Patriarchate also use a black double-headed eagle in a yellow field as their flag or emblem.

Yet to attribute the double-headed eagle motif to Byzantium is erroneous; first, this motif had a multi-cultural history of several millennia before the Byzantines through Rome inherited it; and second, there is absolutely no iconographical or literary evidence that would associate the use of this motif as the official device-flag of the Byzantine Empire.