What were the common exonyms of the Roman Catholic and the Eastern Orthodox churches leading up to and after the Great Schism (1054)?

I think it is correct to say that prior to the great schism there really weren't only two sides in exactly the sense you suggest.

Dating to at least the time of the 1st Ecumenical Council (Nicaea, 325), those Churches which subscribed to the seven Ecumenical Councils were divided administratively into five (5) "Sees": Rome, Constantinople, Alexandria, Antioch, and Jerusalem. This can be seen, if not in the various ancient Church histories available, at least in the Canons of the earlier Ecumenical Councils.

Canon VI of the 1st Ecumenical Council, for example, lays out the foundations of the ancient Patriarchates of Alexandria, Rome, and Antioch (though the term "Patriarch" did not come into use, I think, until the time of Justinian I in the 6th century):

Let the ancient customs in Egypt, Libya and Pentapolis prevail, that the Bishop of Alexandria have jurisdiction in all these, since the like is customary for the Bishop of Rome also. Likewise in Antioch and the other provinces, let the Churches retain their privileges. And this is to be universally understood, that if any one be made bishop without the consent of the Metropolitan, the great Synod has declared that such a man ought not to be a bishop. If, however, two or three bishops shall from natural love of contradiction, oppose the common suffrage of the rest, it being reasonable and in accordance with the ecclesiastical law, then let the choice of the majority prevail.¹

The phrase "Let the ancient customs prevail" indicates that the organization indicated had existed for quite some time up to that point.

Canon VII of the same Council delineated the authority of the Bishop of Jerusalem:

Since custom and ancient tradition have prevailed that the Bishop of Ælia [i.e., Jerusalem] should be honoured, let him, saving its due dignity to the Metropolis, have the next place of honour.²

The place of the last of the five was firmly established at the 2nd Ecumenical Council at Constantinople (381), in Canon III:

The Bishop of Constantinople, however, shall have the prerogative of honour after the Bishop of Rome; because Constantinople is New Rome.³

That is not to say, however, that the eastern Sees did not see themselves as somehow distinct from the See of Rome, which encompassed geographically an enormous area of non-Christian pagans at the time of the earlier Councils. (I have heard it said that at the time of the 1st Ecumenical Council there were about 1,800 bishops in the eastern Sees and only 1,000 bishops within the See of Rome.) Canon V of the 2nd Ecumenical Council refers, for example, to the "Westerns", meaning "Western bishops":

In regard to the tome of the Western [Bishops], we receive those in Antioch [i.e. the Synod of 378] also who confess the unity of the Godhead of the Father, and of the Son, and of the Holy Ghost.⁴

So perhaps the best direct answer to the question, What terms did the Roman and Greek sides of the church use of each other, would be "Easterns" and "Westerns", or perhaps "Eastern Church" and "Western Church".

Although the schism of 1054 is best known, it should be noted perhaps that there were many "mini Schisms" among the five Sees during the first millennium in which one or more sided with Rome against Constantinople. One good example might be the Monothelite controversy of the 7th century, during which both Rome and Jerusalem objected to an innovation led within the See of Constantinople regarding the relationship of the divine and human wills of Christ. Pope Martin, who anathematized those following the doctrine, was eventually arrested by the eastern Emperor, brought to Constantinople, tortured (along with the Byzantine martyr Maximus the Confessor), and eventually exiled to the Crimea, where he was starved to death by barbarians. (The Patriarch of Jerusalem escaped persecution due to the dubious fortune of being under siege by Muslims). It might be interesting to note that in the Synaxarion of the Eastern Orthodox monastery of Simonas Petra on Mt. Athos, the "Catholic" Pope Martin is referred to as a "pillar of Orthodoxy".

The "Great Schism" was primarily a schism between the Pope of Rome and the Patriarch of Constantinople, who mutually anathematized each other. Subsequent to the Schism, however, the remaining three Sees elected to remain in communion with Constantinople rather than Rome. Much of this may have been due to the divergence between the Latin west and Greek east that developed subsequent to the ascension of Charlemagne. Metropolitan Kallistos Ware gives a concise overview of how this occurred in his book, The Orthodox Church:

[In the century following the Apostles], the unity of the Mediterranean world gradually disappeared. The political unity was the first to go. From the end of the third century the Empire, while still theoretically one, was usually divided into two parts, an eastern and a western, each under its own Emperor. Constantine furthered the process of separation by founding a second imperial capital in the east, alongside Old Rome in Italy. Then came the barbarian invasions at the start of the fifth century: apart from Italy, much of which remained within the Empire for some time longer, the west was carved up among barbarian chiefs. The Byzantines never forgot the ideals of Rome under Augustus and Trajan, and still regarded the Empire as in theory universal; but Justinian [I] was the last Emperor who seriously attempted to bridge the gulf between theory and fact, and his conquests in the west were soon abandoned. The political unity of the Greek east and west was destroyed by the barbarian invasions, and never permanently restored.⁵

Metropolitan Kallistos goes on to describe other historical factors leading up to the schism, including:

• Slav and Avar invasions of the Balkan peninsula in the late 6th and 7th centuries, creating a geographical barrier between Constantinople and Rome
• The passage of the Mediterranean into largely Muslim control
• The interim schism between Rome and Constantinople during the Iconoclast controversy in the 8th century, during which Rome took the "Orthodox" position in the matter and which led in part to Rome turning to the Franks for political and military support
• The installation of Charlemagne as Emperor in the west, which was not recognized by the Byzantines.

Regarding the use of the terms "Orthodox" and "Catholic", the terms were used by both western and eastern Church Fathers to refer to the entirety of the Church - east and west. The writings of Popes Leo the Great and Gregory the Great frequently make use of the term "orthodox" (e.g. "He has marked certain places as heretical which are catholicly expressed and entirely orthodox" - Gregory the Great, Epistle XIV⁶). As an eastern example, Athanasius the Great, of Alexandria, refers to the "Catholic Church" in describing the deposition of Arius:

Although you have already subscribed to the letter I addressed to Arius and his fellows, exhorting them to renounce his impiety, and to submit themselves to the sound Catholic Faith, and have shewn your right-mindedness and agreement in the doctrines of the Catholic Church⁷

I am not aware of a single instance nor could I find - at least in those writings preserved in the Nicene Fathers series - any case prior to the Schism where Rome exclusively was referred to as "Catholic" and the eastern Sees (Constantinople, Alexandria, Antioch, and Jerusalem) as "Orthodox".

<sup>1. Nicene and Post-Nicene Fathers, Series 2, Volume 14
2. Ibid.
3. Ibid.
4. Ibid.
5. p.44-45
6. Nicene and Post-Nicene Fathers, Series 2, Volume 12
7. Nicene and Post-Nicene Fathers, Series 2, Volume 4</sup>