Anglicanism involves a series of loyalties, not just to an institution but also to a certain way of understanding and doing things. If one does not grasp that about Anglicanism then one is likely to fall into the same trap as the TAC, which, in its effort to make peace with the Papacy, has got into the habit of treating Anglicanism as nothing more than an ethnic variation of Catholicism - like the Irish or Hispanic Roman Catholic Parish up the street. This reductionist view of Anglicanism is historical hogwash, not to mention theologically dubious.
When the Church of Ireland was disestablished in 1871 it chose four epithets to describe itself - Catholic, Apostolic, Reformed, Protestant. It chose these words because it felt each one of them conveyed an important concept in defining the Church's character, outlook, mission, and purpose. I propose to look at each one of these terms in turn, and draw out a little of their theological and historical meaning.
Catholic - comes from the Greek for "the whole" or "universal." Anglicans have retained both the universally accepted Scriptures - Old Testament, New Testament, and Apocrypha - but also maintained the traditional understanding of how Scripture is to be read. Article 7 makes it clear that not only is the Old Testament not contrary to the new, but that it also preaches Christ. The Articles in accordance with the teaching of St Jerome, also grants a particular status the the Apocrypha - that it is inspired, but not to be used on its own to estabish doctrine. This point of view is consonant with the teaching of both the Eastern and Western Churches from the fourth century down to the Council of Trent when the Roman Church gave the Apocrypha full Canonical Status. The Anglican tradition also accepts the three Catholic Creeds - the Apostles' Creed, the Nicene Creed and the Athanasian Creed. It also retained the universally accepted ministry of Bishop, Presbyter/Priest, and Deacon, and the Dominical Sacraments of Baptism and Holy Communion. At the same time it acknowledges the sacramental character of confirmation, penance, marriage, unction, ordination. All of this is in line with the Church's teaching in the fourth and fifth centuries.
The use of word Apostolic makes two associations in my mind, and both of them were probably in the thoughts of the framers of the Church of Ireland Constitution. The first association is that an Apostolic Church is one founded on the principles of the Apostles, if not by the apostles themselves. This asserts the Anglican claim to have returned to the faith of the primitive Church. The second is that it also reminds us of the doctine of Apostolic Succession in which the transmission of Holy Orders is associated with the preservation of an orthodox (rightly glorifying) Faith in the Blessed Trinity.
The next two terms unfortunately make certain modern Anglicans cringe "Reformed" and "Protestant" but nonetheless, it takes a revisionist historian on something stronger than good port to argue that they are not part of the Anglican heritage.
The first of these "Reformed" was doubtless agreed upon because it reflected a High Church - Low Church consensus that we have since lost. High Churchmen of the period would have seen Anglicanism as reformed when compared to the relatively unreformed Church of Rome. To this Low Churchmen (Evangelicals anyway) would have added their understanding that Anglicanism at least tolerated the tenets of Reformed Christianity including Predestination, Receptionism, and Charitable Presumption attitude to the doctrine of Baptism Regeneration, all of which would have been questioned by Protestant High Churchmen.
Finally, there is the term Protestant. This historically has its origins in the Protestatio of the Evangelical Princes against the Emperor Karl V's attempts to reimpose Catholicism in Germany. It also carries the sense of a Protest in favour of reform, the sufficiency of Scripture and a return to the Christianity of the Patristic Age, which is what both Luther, and the later Anglican Reformers were trying to achieve. Perhaps it is William Van Mildert (1767-1836) who summed up this positive reformation best when he wrote that, "at the Reformation the Church of England became protestant in order to become more truly and fully catholic."
Another thing to which Anglicans adhere is the liturgy. The Book of Common Prayer represents the reformation of the liturgy as surely as the Thirty-nine Articles and the Homilies represent the Reformation of doctrine. The Prayer Book revised and condensed the service of the mediaeval Church with some definite objects in mind. For a start, the liturgy was put into English (and at a later date Welsh, Irish, Gaelic, Jerrais, Manx and a host of other languages) so that worship would be in a language "understanded of the people." It also removed Prayers to the Saints - whilst acknowledging the Communion of Saints through the Calendar - and also removed all legendary material. Just from this rather casual enumeration of the principles behind the Prayer Book one can see why Bishop Manning of New York and Bishop Nichols of California both slammed the relatively moderate American Missal as "misrepresenting" the Prayer Book. It is also notable that until the 1960s English clergymen promised to use the official liturgy of the Church - the 1662 BCP, and the 1928 BCP is part of the doctrine, disciple and worship of the Church which American clergymen give their oath to uphold.
It is absolutely no coincidence that the Thirty-Nine Articles are placed in the back of the Book of Common Prayer. The two of them together are a sort of Anglican "here I stand." A repudiation of either is really a repudiation of Anglicanism itself. I believe that for Anglicanism to prosper in the twenty-first century it is neccessary to return to our Anglican heritage, and that without dissimulation or special pleading.