Though I can't speak for the classical conception of the city of Rome's place in imperial ideology, it had an interesting place in that of the middle Byzantines. Even before the loss of Rome in the mid 8th century, the great lengths that Constantine went to portray New Rome as a valid successor to, and even continuation of, the old city were very much present in Byzantine thought. Constantine had transferred many members of the old senate to that of New Rome, founded it on seven hills, given it suitable entertainment, and provided it, though much more of this was done by his successors, with the forums, monuments, sewers, and baths characteristic of a great Roman city. New Rome, often called "the new Rome"(1), or sometimes "Our Rome"(2) in Byzantine literature, was very much characterized as the spiritual continuation of the eternal city, which is how the Byzantines justified their lack of control over their perceived city of origin whenever the question came up(3).

(1) The 9th century Patriarchal seal of Photios and the epitaph or Basil II

(2) From Michael Psellos's Chronographia

(3) See Luitprand of Cremona's Report on embassy to Constantinople, for example.