Is Independent Catholicism even Catholic?
Yes, definitely. There are several expressions of Catholicism, and many Catholic churches comprise the "one, holy, catholic and apostolic Church" that is professed by all.
In the Western world, the largest and most well-known expression of Catholicism is the Roman Catholic Church. In Asia and Eastern Europe, however, the most well-known expression of Catholicism is the Orthodox Catholic Church, which possesses over 300 million adherents and which excommunicated the Roman Catholic Church in 1054 A.D. for its deviance from the ancient creed of the Church, its insistence on unleavened bread, and its imposition of the novelty of clerical celibacy.
What do all Catholic churches have in common?
All Catholic churches, and indeed most Christian churches, profess faith in the "one, holy, catholic and apostolic Church."
Relating with one another as sister churches, these members of the universal Church believe that by baptism, we become members of the one Body of Christ and are nourished by the eucharist. The apostolic faith that has been handed down to us also teaches that there are other sacraments that we celebrate during important moments in life.
Like sisters in the same family, these churches share various similarities, and yet they are all different. As sister churches, most of our differences are administrative and disciplinary, but some of our differences are ecclesiological and theological. Differences are to be expected; not all sisters are the same.
Still, we are united by those bonds that we all hold dear: baptism, eucharist, and apostolic succession.
What does it mean to be one?
We often use a capital "C" when we refer to the universal Church (viz., the Body of Christ, which is composed of all sister churches), and with humility we use a lowercase "c" to refer to the many, particular churches to which we belong (unless used as a proper noun).
Despite any divisions that might exist, we recognize that all our sister churches belong to the one, universal Catholic Church. As such, despite our differences, we all form part of the People of God. Christ is the vine, and we are the branches (Jn 15:5). Christ is the head of the body, the Church (Col 1:18), of which we are all members (1Cor 12:12-27). We celebrate our unity (Gal 3:28), and we recognize our diversity (Rom 12:4-8).
What does it mean to be holy?
As individuals, we are simultaneously sinners and saints. We know that in our history, the same can be said of the institutions that are comprised of human beings. In the name of supposed good, our institutions have historically upheld narrow views, hosted inquisitions, waged crusades, advocated for slavery and the repression of various voices, and eliminated entire cultures. Despite this, we know that the work of redemption is somehow being carried out in us. Even though we sometimes stumble and get off track (Lk 15:13-17), we ultimately desire to keep turning toward that loving and compassionate mystery, which is God (Lk 15:20-24).
What does it mean to be catholic?
According to the Greek root of the word, to be "καθολιχοζ" is to be "whole" and "complete.” Yes, as St. Irenaeus suggested, every Catholic community is whole and complete in itself! In a cosmological sense, we also interpret this word as meaning "universal." We use the word "catholic" of our Church because we believe that it is completely and wholly that which God intends for God's Holy People.
No single branch (or church) within the universal Church can claim exclusive rights to the name "catholic," and no single branch can pretend to be the Vine, from which other branches have been "cut off" (Jn 15:5-6). Christ alone is the Vine; all authentic branches partake of the meaning and challenge of being part of the Vine, and all bear great fruit (Jn 15:7-8).
Humility demands that we continually reflect on the deep significance of Christ's words: "I have other sheep that are not of this flock. Them, too, I must bring, and they shall hear my voice, and there shall be one flock and one shepherd" (Jn 10:16).
What does it mean to be apostolic?
By word and deed, we cling to and stir into flame the faith handed down to us by the apostles (2Tim 1:6). These apostles planted various churches as they preached the gospel throughout the world. Though these churches have varied in expression throughout the ages, depending on the context in which they sprang up, they all trace their lineage to the apostles. Some of these local churches eventually yielded to Roman Catholic or Orthodox Catholic authority, while others continued to strive to live out the authentic message of Christ in their own way. By tracing their apostolic succession--their lineage back to the apostles--these churches confirm that they continue to pass on the gift of the Spirit that was received by validly consecrated ministers who, in turn, share the gift of that Spirit with other ministers.