The symbol is known as a Chi-Rho and is a Christogram, a symbolic shorthand for Jesus Christ used either as a symbol of faith or as an abbreviation for "Jesus Christ" in a text. The symbol is merely an amalgamation of the Greek letters Chi (X) and Rho (P) which are the first two letters of Christ when written in Greek. A modern equivalent might be somebody referring to Jesus as J.C.

The symbol doesn't 'stand' for either the Western or Eastern Empires, per se; neither is the 'real one'; the Chi-Rho is a sign of Christian faith that was then coopted post-Constantine as a sign of Christianity becoming the Imperial faith. Eusebius recounts that Constantine carried a Chi-Rho standard into battle, as a sign of his faith and divine support, for example. The symbol pops up in a wide array of archaeological contexts across both Empires: it appears on helmets, cathedrals, edicts, a number of domestic decorations found in villas in Britain even. Think of it not as a political symbol per se, but something like how a crescent appears on the flags of Turkey, Tunisia, Algeria, Libya, Pakistan etc.