One expert I consulted, who appears not to believe in reincarnation, made an interesting point. Quoting Origen from the same passage in "Contra Celsum" (1.32) as used by Cranston and Head in "Reincarnation: The Phoenix Fire Mystery," this expert adds the word "Or" to the front of the quote, which potentially changes it, as follows: "Or is it not more in conformity with reason, that every soul, for certain mysterious reasons (I speak now according to the opinion of Pythagoras, and Plato, and Empedocles, whom Celsus frequently names), is introduced into a body, and introduced according to its deserts and former actions? It is probable, therefore, that this soul also, which conferred more benefit by its residence in the flesh than that of many men (to avoid prejudice, I do not say "all"), stood in need of a body not only superior to others, but invested with all excellent qualities." The passage quoted in the Cranston and Head reference begins, "Is it not more in conformity..." The addition of the word "Or" leaves open the possibility that Origen was citing this belief as an alternative by way of representing something he may or may not have subscribed to. I would have to see a larger portion of the argument surrounding that quote to be sure, although the phrase that follows, "It is probable, therefore..." suggests that Origen was, in fact, taking the reincarnationist view.
From a webpage about the "Contra Celsum" itself, we learn that: "This work is a paragraph by paragraph demolition of the lost anti-Christian pamphlet 'The True Doctrine' (Alhqej logoj) by the philosopher Celsus. Celsus wrote around 178 AD, but the work did not come into Origen's hands until the early third century. Origen questions whether anyone will ever have seen the pamphlet--he had not himself before now--but at the request of a friend he has refuted it." This indicates that, indeed, Origen could have been citing something he disagreed with and was in the process of writing against. I have not yet found the original passage.
A brief look into the history surrounding the Fifth Council suggests that the entire matter was highly complex and intensely political. The Catholic source this comment is connected to says that Origen was not even the subject of this Council at all, but instead, it focused on something called the "Three Chapters."
However, take a look at the two websites links below this Catholic one, which cite another, less ambiguous quote* by Origen in "De Principalis," and explain that Origen was not mentioned in the official Fifth Council conclusions which the Pope had to sign off on, but rather (as I read it) as an unofficial act within the council that the Pope was boycotting. It was this unofficial document, targeting Origin's teachings, which yielded the oft-quoted anathema, "Whosoever teaches the doctrine of a supposed pre-birth existence of the soul, and speaks of a monstrous restoration of this, is cursed."
If this interpretation of history is correct, reincarnation was never officially banned in the first place--it was just a political ruse by Emperor Justinian, perhaps as influenced by his wife Theodora, who was a Monophysite. (Warning--popups with these external links.)
In a more recent visit to the Catholic page, at the very bottom of the page I see a clear outline of what is known of the history of the 5th Council and Origen's "anathema". It parallels what is given on the two reincarnationist websites--whether I missed it on my first visit, or whether it's been added, I don't know. All appear to agree that the "Thirteen Anathemata" pronounced against Origen were unofficial, in that they were never signed by the Pope, who was boycotting the meeting called by the Emperor.
*"Each soul enters the world strengthened by the victories or weakened by the defects of its past lives. Its place in this world is determined by past virtues and shortcomings." De Principalis.