The Church's long standing belief is that on June 29, the year usually given as A.D. 67, St. Peter was crucified in Rome on an inverted cross, as he believed that he was not worthy to die the exact same way as Christ. On the exact same day, St. Paul was beheaded for the same faith that St. Pope Peter professed.
This feast in my opinion is a hallmark of the Church's catholicity. But what exactly does the word "catholic" mean?
1. broad or wide-ranging in tastes, interests, or the like; having sympathies with all; broad-minded; liberal
2. universal in extent; involving all; of interest to all
3. pertaining to the whole Christian body or church
The word "liberal" here has a different meaning than it is used to mean today, that is, open to various things. That is exactly what the word is implying, and let me explain why the Church is called such.
The old God-revealed religion was Judaism. It was a very ethnic religion. It was a religion of a people, of a race. On the feast of Shavuot, what the Greeks called Pentecost, the Jewish nation was officially established at Mt Sinai. (I will point out it was originally a harvest festival. It is this background that the Eastern Churches use green on Pentecost, with plenty of greenery.)
On the Pentecost after the Resurrection, the Holy Ghost came upon the Apostles and the Church was established, like the Judaism of old was. As the Church grew, we have this quote from St. Ignatius of Antioch: "Where the bishop is, let the multitude of believers be; even as where Jesus is, there is the Catholic Church." That was written in the first century.
So why that description? Because it is a direct antithesis to the Judaism of old: what was a very exclusive religion became religion inclusive of all people, so that all people could be saved, not just Jews. The traditional translation of "catholic" was "universal": the Church was open to all, not just Jews.
When Sts. Peter and Paul when to Rome, two ethnic Jews, they established the faith there that became the faith of what was once a pagan Empire. These people were not Jews. St. Peter became bishop of a people that owned the land his race lived on.
In like manner, St. Paul, the greatest preacher ever seen, went to various places preaching the Gospel to people who were not Jewish. We have Churches with every name, besides Jewish:
Coptic Church, Byzantine Church, Melkite Church, Maronite Church, Roman Church, Romanian Church, Armenian Church, Syro-Malabar Church, Syro-Malankara Church, etc.
So that is my "sermon" for today's feast. I don't think it is too hard to believe that these two great saints died on the same day. Perhaps that's just what the Romans wanted: to kill all teh major Christian leaders in Rome at the same time. But we know the saying: the blood of martyrs is the seed of Christians.