Technically, Rome IOTL is divided between Italy and the Vatican. That division predates the Cold War -- but if Italy was divided like Germany, and Rome was divided like Berlin, it'd be interesting to see how Rome would be divided.
Assuming a Communist North/NATO-aligned South divide in Italy, would it make sense to assume the Vatican would remain independent? Would it be as dependent on South Italy as in OTL, or would Communism in places like Italy, Poland, Czechoslovakia, and Hungary have a more Catholic flavour? Perhaps Liberation Theology would get some official support from Church leadership -- or at least not be actively condemned by the Papacy?
There were a lot of Soviet-aligned or generally-Leftist religious leaders -- for example, the intersection of political Islam and Marxism in the Algerian War or the Arab-Israeli Wars; Hindu and Buddhist socialists in places like India or Vietnam; and Protestant leaders like Desmond Tutu or Martin Luther King. And a lot of these movements cooperated across sectarian lines -- for example, Malcolm X and Martin Luther King cooperated (and conflicted) during the American Civil Rights Movement; and Muammar Qaddafi offered training and supplies to the Provisional IRA and FARC alongside the PLO, the PKK, and other largely-Muslim Leftist movements. Whatever the USSR's policy towards religion within its own borders, I am sure it'd influence the Papacy if it could, and maybe cultivate Leftist Catholics like Oscar Romero. It helps that the Catholic Church has very limited influence in Russia in general, so whatever happens in Rome, it needn't affect the people of Moscow.