Excellent point!
Although it is not widely known in our Western world, the Catholic Church is actually a communion of Churches. According to the Constitution on the Church of the Second Vatican Council, Lumen Gentium, the Catholic Church is understood to be "a corporate body of Churches," united with the Pope of Rome, who serves as the guardian of unity (LG, no. 23). At present there are 22 Churches that comprise the Catholic Church. The new Code of Canon Law, promulgated by Pope John Paul II, uses the phrase "autonomous ritual Churches" to describe these various Churches (canon 112). Each Church has its own hierarchy, spirituality, and theological perspective. Because of the particularities of history, there is only one Western Catholic Church, while there are 21 Eastern Catholic Churches. The Western Church, known officially as the Latin Church, is the largest of the Catholic Churches. It is immediately subject to the Roman Pontiff as Patriarch of the West. The Eastern Catholic Churches are each led by a Patriarch, Major Archbishop, or Metropolitan, who governs their Church together with a synod of bishops. Through the Congregation for Oriental Churches, the Roman Pontiff works to assure the health and well-being of the Eastern Catholic Churches.
While this diversity within the one Catholic Church can appear confusing at first, it in no way compromises the Church's unity. In a certain sense, it is a reflection of the mystery of the Trinity. Just as God is three Persons, yet one God, so the Church is 22 Churches, yet one Church.
The Catechism of the Catholic Church summarizes this nicely:
"From the beginning, this one Church has been marked by a great diversity which comes from both the variety of God's gifts and the diversity of those who receive them... Holding a rightful place in the communion of the Church there are also particular Churches that retain their own traditions. The great richness of such diversity is not opposed to the Church's unity" (CCC no. 814).
Although there are 22 Churches, there are only eight "Rites" that are used among them. A Rite is a "liturgical, theological, spiritual and disciplinary patrimony," (Code of Canons of the Eastern Churches, canon 28). "Rite" best refers to the liturgical and disciplinary traditions used in celebrating the sacraments. Many Eastern Catholic Churches use the same Rite, although they are distinct autonomous Churches. For example, the Ukrainian Catholic Church and the Melkite Catholic Church are distinct Churches with their own hierarchies. Yet they both use the Byzantine Rite.
To learn more about the "two lungs" of the Catholic Church, visit this link:
The Vatican II Council declared that "all should realize it is of supreme importance to understand, venerate, preserve, and foster the exceedingly rich liturgical and spiritual heritage of the Eastern churches, in order faithfully to preserve the fullness of Christian tradition" (Unitatis Redintegrato, 15).
A Roman rite Catholic may attend any Eastern Catholic Liturgy and fulfill his of her obligations at any Eastern Catholic Parish. A Roman rite Catholic may join any Eastern Catholic Parish and receive any sacrament from an Eastern Catholic priest, since all belong to the Catholic Church as a whole. I am a Roman Catholic practicing my faith at a Maronite Catholic Church. Like the Chaldeans, the Maronites retain Aramaic for the Consecration. It is as close as one comes to being at the Last Supper.