I would like to second all of Diak’s wonderful words and add just a few of my own about what “communion with, not submission to” means to me.
Sometimes it seems to me that some Latins are uncomfortable with this expression, because it may appear to indicate a less-than-firm relationship with the Pope. Personally I think it represents a much more firm relationship than “submission” ever could. Anyone can simply submit while disapproving in their heart, but to stand with someone in true communion is an act of real devotion. Sometimes there are rough spots, but historically when most Eastern Catholics have said, effectively if not in these words, “I am with Peter and the Pope”, they’ve meant that they were willing to die for the unity of the Church, and that they fully accepted the Petrine role of the Papacy while at the same time maintaining the dignity of their own Apostolic traditions (some of which also come down through Peter in different areas, such as Antioch, incidentally). Many of these people
have died for this communion, and in the case of the Ukrainian Catholic Church (which is often at the forefront of this particular mindset we are discussing) it has been all but completely absorbed and disbanded by the Russian Orthodox only to be revived out of whole cloth spontaneously by the devotion of the Ukrainian Catholic people.
“Communion with” means accepting that Rome has a special role in the Church, which includes the heavy and unenviable burden of at times arbitrating for all Catholics from the standpoint of preserving our universal Tradition. Even then, however, it is not submission, but a “standing with the Pope”; he supports Tradition, and we uphold the bond of love, in love. He is not some authority figure that lords over the Catholic Communion, but an “elder brother” tasked with a particularily heavy cross on behalf of all of us. Sometimes we must remind him (and others) of the intererests and traditions of the non-Latin Churches, but this is not merely a hard-headed stance, but also one of defending and supporting the whole Catholic Tradition (in line with Bp. Sheen’s quote above).
So I would hope that every Catholic can say “Communion, not submission”, it’s a strong bond that does nothing to lessen the authority of the Church, and everything to highlight the fact that we are a family of Love who stand together in Faith and Hope.
Peace and God bless!