I just read an interesting essay on how GRRM uses the names that PoV characters give to non-PoV characters to both illustrate relationships and influence [or perhaps manipulate] readers' feelings toward that character. This can be powerful, and it also can be misleading, since many of the PoV characters are biased or in other ways unreliable.
But this tool has other uses. It can also show how a PoV character perceives himself. One of the many plot complexities of ASoIaF is the recovery - or perhaps redemptive - arc of certain characters. Arya, Sansa, Jaime, Tyrion and Theon all fit this trope, each in his/her own way*.
One thing that struck me on a reread of ADwD was how Theon is presented in his own PoV chapters. Theon has what must be the deepest identity conflict** of any character in the saga, a point often reinforced with internal asides such as: I must remember my name. Reek - it rhymes with meek. But somehow Reek is eventually set aside, and Theon re-emerges - a Theon with enough reckless courage to rescue and escape with the faux Arya.
Before the rise can begin, Reek has to reach bottom - an absolute nadir of physical and emotional humiliation. This happens in the chapter The Prince of Winterfell. Notably, at the beginning of the chapter, he is Theon Greyjoy, as he drapes a fur-trimmed cloak about the shoulders of Ramsey's bride to be. At the end of the chapter, after the wedding ceremony, Ramsey commands him to prepare his new bride for the bedding, in a most intimate lingual way, and we read this.
Somewhere in the godswood, a raven screamed. The dagger was still in his hand.
He sheathed it.
Reek, my name is Reek, it rhymes with weak.
Reek bent to his task.
Setting aside the dagger symbolism, we see here the utter hopelessness and subjugation of Reek. After his betrayal of the Ironborn at Moat Cailin, he fell into a deep pit of worthlessness and despair, and this is the nadir.
Four chapters later, Theon appears again in The Turncloak, initially as Theon Greyjoy, and we can see the transformation begin. He tells Jeyne, "Just be Arya," and admonishes her that: "In here I'm Reek." This is the first sign that Reek is starting to fade from an identity to a facade. He then muses on his fate to come when Ramsay inevitably tires of his new plaything. He doesn't state this out right, but he comes to the realization that his fate is sealed, no matter what he does. It's only a matter of time before his flaying is completed. So at that point he also realizes that he has nothing left to lose.
When Rowan approaches him, saying, "Theon Greyjoy," We read, "My name is Reek, he almost said," another subtle sign that he is beginning to separate himself from Reek. They discuss other ways in which he might be remembered: Theon Turncloak, Theon the Clever. He rebuffs her, suspecting it is another of Ramsay's ploys. "Reek, Reek, my name is Reek. I must not forget my name," he thinks, but this time it's ambiguous and conflicted. He then wanders aimlessly around Winterfell, eventually ending up in the godswood. And there he prays, lost, confused, alone, and has his epiphany. When he returns to the great hall, it is as Theon Greyjoy.
These are small subtle things, easily missed, but important signs in GRRM's method of story telling. This is as far as I gotten with Theon in this reread. I strongly suspect we will never see him think of himself as Reek again.
* I did not mention Sandor Clegane in this context because I think of his arc differently. These others all fall from a condition of relative comfort into deep travail, then struggle to find a way out, which is transformative. Sandor is at rock bottom when we first meet him, and has been so since childhood. Corseque examines his transformative arc here.
** Or possibly Arya. But her conflict is less internal, and I think she always knows that she is Arya of House Stark.