Both. both Romes as Christian Empires used this symbol. Two heads, one for Church one for State and one claw holds a sword while another holds a cross.
Although I have seen the explanation above before, I have also read that the Double-Headed Eagle was adopted by Emperorr Justinian in the sixth century when his General Bellisarion successfully recovered Rome, Italy, and much of the rest of the West from the Visigoths restoring one unified Empire (it had been broken in two earlier with two separate emporers). The two heads - one facing West toward Rome, the other East toward Constantinople- denoted the union. Other symbols included the cross for the Orthodox Church and the Sword for the Emperor. After the fall of Constantinople the Russian Tsar adopted the Byzantine Eagle to symbolize Russia as the "Third Rome".
An interesting aside comes from my ancestral area of Asia Minor -Trebizond- where a Greek (Orthodox) Empire continued after the fall of "the City" from 1453 to 1461. The Empire of Trebizond adopted a single headed eagle with the head facing west towards both Rome and Constantinople. The Pontic Greeks today still use this symbol.
History is great, isn't it!