Catholic means universal, so both present themselves as the original and true church, with the head either in Rome or Constantinople/Pentarchy. The actual break of communion comes from 1054 but really began much earlier.
Even in protestant churches like the Presbyterians and the Methodists you will hear references to the "Catholic Church" where it refers to the universal church that is inclusive all all believers regardless of denomination. For example in shows up in the Nicene Creed and the Apostles Creed.