Indeed, the Roman Rite has traditionally always had an robust system for determining which feast day should take precedence. Before the 1969 reforms, this had feasts with different classifications, such as doubles, semi-doubles and so on.
The Eastern Orthodox and Byzantine Rite Catholics also have this, in that the Feasts of our Lord have the highest priority, and then we group the other holy days into those pertaining to the Theotokos (two of which overlap with Dominical feasts, namely Candlemas and the Annunication), of the Holy Apostles, of the Holy Cross (such as the third Sunday in Lent and the Exaltation of the Cross on the 14th of September), and of martyrs, confessors, holy hierarchs and ascetics, and these along with the time of year influence the liturgical colors in the Orthodox churches, albeit in different ways (the Typikon only specifies whether light or dark vestments are to be used, and so for those churches that have an organized system of liturgical color, it amounts to a jurisdictional, diocesan or even parish-based tradition, with variations including, for example, whether to use white vestments only on Pascha after changing out of purple, crimson or black (another decision to be made) on Holy Saturday, or whether after Paschal Matins to switch to red vestments, and this variation exists even within the Russian Orthodox (ROCOR tends to use white vestments only, whereas the OCA and MP tend to use white and then red, but this is not universal and really depends on the parish). Green vestments are traditionally used on Palm Sunday, Pentecost and All Saints Day (the Sunday after Pentecost, since Pentecost is both the feast of the decent of the Holy Spirit and the feast of the Trinity in the Byzantine Rite, whereas in the Roman Rite the two were separated into two different feasts, much like how in all churches except the Armenian church the Nativity and Epiphany or Baptism of Christ were separated into two feasts in the fourth century), but are also used in some churches for the feasts of confessors and ascetics, but amusingly green is often very seldom used, being used in some churches only as frequently as the Roman Rite uses rose vestments. In most churches the default liturgical color is gold.
However, the Byzantine Rite, being deserving of the name Byzantine for its complexity, also features fairly elaborate rubrics for celebrating two events when they converge. A complete liturgical library in the Orthodox Church might include around 20 folio-sized books, such as the books for each tone of the Octoechos, that constitute the main hymnal, and then the Monthly Menaion which contains the propers for each day in the month, and the shorter summary Festal Menaion, and the General Menaion which one can fall back on for a feast if one doesn’t have rubrics specifically for it, and then the Triodion, which contains the hymns and services for pre-Lent, Lent and Holy Week, up to the Vesperal Divine Liturgy on Holy Saturday, which is extremely similiar to the pre-1955 Roman Rite Paschal Vigil Mass, which before 1955 featured 12 Old Testament prophecies, which like the 14 of the Byzantine Rite, were read while baptisms were performed of catechumens, and which was celebrated in the morning, like the Byzantine Rite equivalent, although there were some differences, for example, in Slavic Orthodox churches the black vestments and paraments of Good Friday are removed at the Vesperal Divine Liturgy and replaced with white, for Paschal Matins, as mentioned above.
At any rate, we have all these liturgical books, but there is one book that controls everything, the Typikon, and it can vary between jurisdictions, between monasteries (since in a monastery it will also include the monastic rule, so it functions both as a liturgical Ordo and as something like the Rule of St. Benedict, which also included liturgical instructions but not to this extent), and even between parishes. But regardless of which typikon one is using (the two most common styles being those based on the Studite-Sabaite typikon, used on Mount Athos and in most Slavic churches, and the Violakis typikon, used by most Greek and Arabic churches, which reflects common practices in those parishes in the 19th century. There is also the Cathedral Typikon, which is no longer in use but which has been reconstructed and its services recorded by Dr. Alexander Lingas, the brilliant leader of the choir Capella Romana, who has also recorded several albums of interest specifically to Eastern Catholics, for example, several recordings of places where the East meets the West in terms of church music, such as on Cyprus, where the Roman Catholic churches used something close to Byzantine chant and the Orthodox something closer to Gregorian chant, although sadly he has not yet done Milan or Toledo - Ambrosian chant and Mozarabic Chant are often remarked upon for being like a fusion of Eastern and Western liturgical sounds.
Much of the Typikon, in any case, pertains to what to do when two feasts fall on the same day, and in many cases, where the feasts are particularly important, such as Pascha and the Annunciation, they are combined. If one uses a typical Studite-Sabaite typikon, because of the different liturgical cycles such as the eight week cycle of the Octoechos, the fixed feasts, the movable feasts, and the eleven Matins Gospels cycle (eleven lections describing the post-resurrection appearances of our Lord, read at Matins except when there is a proper Gospel to be read instead, which is rare), even though all the hymns are pre-selected, no liturgy will be exactly the same in terms of prescribed services for at least 537 years, because of the variations caused by the interactions of these different systems.
