[Note: This post originally made a slightly stronger case, but upon further research, I have modified the argument somewhat.]
There is a consensus in scholarship, which until yesterday I assumed to be certain fact, that, whereas earlier generations had seen a variety of poems ascribed to Orpheus arise and disappear, or being reworked again and again, the late Neoplatonists of the 5th and 6th centuries CE were looking to a single canonized epic, the Sacred Discourses in 24 Rhapsodies. But in fact, there are hardly any certain facts about this work. This precise title is given only in the Suda, a 9th-century encyclopedia, without any indication of their contents. Other references to Orphic works by the phrase “sacred discourses ”(hieroi logoi), or even just “discourses” (logoi), are altogether too vague to treat as formal titles, and are not connected with the fragments of the Neoplatonists anyway. Further, only one author refers to a numbered rhapsody, namely the so-called Tübingen Theosophy 61, which attributes two lines about Phanes to “the fourth rhapsody to Musaeus”; but note that even this implies a different title, namely Orpheus to Musaeus in X Rhapsodies. If both of these titles refer to the same work, then we should rather say that it has no formal title at all.
The only connection of the shadowy Orphicum of the Suda and the Tübingen Theosophy to the Neoplatonists is that Damascius refers to Orphic rhapsodies three times in one passage of his great treatise On Principles. This, however, is no connection at all, unless it could be shown either that, in the Orphic arena, the word rhapsody was the exclusive property of the Sacred Discourses and/or Orpheus to Musaeus, or that the Neoplatonists quote from Orpheus in a way that suggests there was one work consisting of a set number of rhapsodies in a fixed order, like the Tübingen Theosophy does. But neither is the case.
Because of this, and because neither Damascius nor other Neoplatonists use the word rhapsody to refer to Orphic works in any other context, a much less reckless theory is that Damascius was grasping for words to refer specifically to the contents of the Orphic poetry he knew, and which was in use among the Neoplatonists, as opposed to the divergent summaries of Orphic theology that he found in older prose works, because he was discussing both in said passage (namely On Principles p. 316-320). The terms he lands upon are “the theology in those Orphic rhapsodies that are in circulation” (316), “the accustomed Orphic theology” (317), “the theology in the rhapsodies” (318), and “Orpheus in the rhapsodies” (320). Everywhere else when Damascius has the same body of poetry in mind, he refers simply to Orpheus, a few times to “the theologue” or, in one instance each, to “the Orphica”, “Orpheus the theologue”, “the Orphic theologies”, and “the Orphic theology” (mentioned alongside the Chaldaic and Egyptian theology and that of the Phoenicians in On Principles p. 219). Either, then, Damascius fastens upon an incidental quality of the Orphic poems he knew (that one or more of them were subdivided into rhapsodies, or could be described as rhapsodies); or he considers Rhapsodies (not Sacred Discourses in 24 Rhapsodies!) as the title, but strongly prefers not to use it.
Syrianus, the teacher of Proclus, and Hermias, another student of his, refer only to Orpheus and do not name any poem of his; the same is true of Olympiodorus and the Christian John Philoponus (both students of Hermias’s son Ammonius), as well as Simplicius, the student of Damascius. If we had only these authors, we might not even think that an Orphic poem would have a title (and perhaps we would be right).
Really, it comes down to Proclus, as his works are the most extensive and most studded with quotations of Orphica. He again often refers simply to Orpheus or Orphica (often indistinguishable from Orphics qua persons, who also occur), as well as “Orpheus the theologue”, and often merely “the theologue” (also “the theologue of the Hellenes”; when Proclus cites “the theologues”, on the other hand, this can mean many different things and is often deliberately general or vague, and only rarely means specifically Orpheus/Orphica).
When referring to the Orphic texts and/or their contents, Proclus calls them by an embarrassment of names: “the Orphic writings”, “the Orphic verses”, “the Orphic tradition(s)”,* “the Orphic myths”, “the Orphic theology”, “the Orphic theology of the Hellenes”, “the theology of the Orphica (or Orphics?) which is the same as the Hellenic”, “the Orphic theomythia”, “the Orphic theologies”, “the Orphic theogonies”, and “the Orphic genealogies”. These are all fascinating appellations, but what they are not is anything like a formal title; in fact the variety strongly suggests there was no title, at least not in Proclus’s copies. If there was, we have no reason to think it was Sacred Discourses or Rhapsodies, let alone Sacred Discourses in 24 Rhapsodies.
(*Note that traditions here refers to things handed down in writing, not to a tradition of practice, initiation or oral teaching.)
In sum, we can say no more than that the Neoplatonists were working with what they saw as a coherent body of Orphic poetry, which they usually cited by naming Orpheus, but also (without difference in reference, it seems) attributed to the Orphics. They referred to it in various singular and plural terms, suggesting that more than one poem or narrative were in play (and perhaps multiple tellings of the same narratives, since Proclus mentions “theogonies”, plural); further uncertainty arises from the fact that they preferred to speak of it as a theology, a word that can mean a text but also the coherent ideas contained in a plurality of texts.
The only Orphic poem which a Neoplatonist actually mentions by name is the Orphic Hymn to Number, which however is also called the Pythagorean Hymn to Number, and thus is clearly an exceptional case to begin with.
If, despite all this, the scholarly consensus should in fact be right in that the title in the Suda, Sacred Discourses in 24 Rhapsodies, refers to essentially the same collection used by Proclus and his fellow late Neoplatonists (and that is a big If), then still, (a) we have to conclude that neither this nor any of the other appellations was a fixed title, (b) there is very little to corroborate that there was a fixed sequential order of rhapsodies,* (c) there is nothing at all to corroborate that the number of rhapsodies was consistently fixed at 24, and (d) a compilation of rhapsodies for which we cannot prove that name, number or order were consistent should not be treated as a canonized and closed text: lines, passages and even whole rhapsodies may very easily have been in fluctuation even if the notion of a single body of Orphic rhapsodies was as widely shared as has been assumed (an assumption, I will stress again, which rests on very shoddy foundations).
[*The single piece of corroboration, to be clear, is in the Tübingen Theosophy, as mentioned above. Running counter to this, several fragments that have been assigned to the Sacred Discourses in 24 Rhapsodies by editors are cited according to topic rather than number: On Zeus and Kore (fragment 287 Bernabé/115 Kern), The Discourses about Hipta (fragment 329 Bernabé/199 + 211 Kern), The Destruction of Dionysus (fragment 330 Bernabé/206 Kern)]
ADDENDUM: Some fragments that have been classified as Orphic because Proclus ascribes them to the theologues probably should be excluded from the category. In particular, Proclus says that the two lines “Wretched humankind is your tears, / but when you laughed, you sprouted forth sacred godkind” are by “the theologues” or “some person hymning the Sun”, in other words: are from an anonymous Hymn to the Sun.
ADDENDUM 2: I initially overlooked Marinus’s Life of Proclus, which contains some further mentions of Orphic texts/traditions, including one reference to rhapsodies. Firstly, “Orphic verses” are mentioned alongside hymns as texts that Proclus had memorized (ch. 20); then “Orphic theology” is discussed as something of which one can teach the “rudiments (stoikheia)” without teaching “the verses” (ch. 26); Syrianus wrote “commentaries on Orpheus” (ibid.); Proclus gave lectures on “the (works) of Orpheus” (in Greek, τὰ Ὀρφέως), which Marinus calls ποίησις, a word that can equally mean poetry in the abstract or a long work of epic poetry (ch. 27). Finally, in the same chapter, “Orpheus” is equivocated with “the entire theomythia or all the rhapsodies”, yet another indication of the ambiguously singular/plural character of the Orphic corpus of the Neoplatonists and the absence of a fixed or formal title. My prior conclusion, that we cannot say whether the Orphica of the Neoplatonists were one or several texts, holds, but Marinus does imply that for the circle of Proclus, there was a closed canon of some sort, such that a commentary on its entirety was imaginable.
ADDENDUM 3: It is possible that the newly discovered palimpsest containing hexameter poetry about the childhood of Dionysus belongs to the corpus used by the Neoplatonists, and also possible that it is from the Sacred Discourses; it is possible, again, that it is both, but caution is in order. If the Dionysiaca of Nonnus were lost and a few pages about the childhood of Dionysus were recovered, it may well have ended up being ascribed to this phantom of scholarship as well. Even if the editor’s conclusion is correct in that the new poem is Orphic, it would take extraordinary evidence to connect it with a specific text or other group of fragments. Orphic poetry, after all, is infamous for its textual instability and the fact that it continued to be reworked repeatedly over antiquity.
FINAL NOTE: The Late Neoplatonists knew at least a few hymns ascribed to Orpheus, but they show no awareness of the now prominent corpus of Orphic Hymns. That said, there is a degree of overlap between the contents of their “Orphic theology” and the ideas contained in the Hymns.