Where does the peripeteia (reversal) occur in the play Oedipus Rex?
Both the recognition and reversal occur when it is revealed that he was adopted. This is a key piece of the puzzle.
First Oedipus recognises that he is the man he has been looking for and as a result we have reversal. Everything he had known or believed about himself and his life turned out to be a lie. He was not who he thought he was.
Not only did he kill his father when he was a young man trying to escape his fate, but he unknowingly married his mother and had four children with her. His children were also his half brothers and sisters.
As you can see, the peripeteia, did indeed "reverse" his life.
Shaketeach is correct, but there is a specific moment that can be pinpointed as the peripeteia. The Messenger who comes to tell Oedipus that the man he believed to be his father, Polybus, is dead and that he will be made king in Corinth, also happens to be the man who brought the baby Oedipus to King Polybus and his wife.
The scene that reveals Oedipus' true past goes on for quite a few pages, drawing out the suspense as Oedipus learns that, according to the Messenger, he need not be afraid of murder and incest in Corinth since Polybus was not his natural father.
And yet the unravelling doesn't end with this news. Oedipus demands to have the old Herdsman who gave him away to the Messenger brought forward. He says:
With such clues,
I could not fail to bring my birth to light. (1058-9)
After more suspenseful questioning, it is finally revealed that the Herdsman got the child from Jocasta and gave it to the Messenger. Finally, Oedipus has his answer. He says (lines 1181 - 1185):
O, O, O, they will all come,
all come out clearly! Light of the sun, let me
look upon you no more after today!
I who first saw the light bred of a match
accursed, and accursed in my living
with them I lived with, cursed in my killing.
This is the exact moment of reversal.
By reversal he basically re evaluates his life. Once he comes to term that yes he is the killer of his father and husband of his mother, he goes over in his head events that led to this and why this happened. He becomes disgusted with himself and the fact that he even bore children, which are in fact his brother sisters/children. His reversal is his re evaluation of what has happened.