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The total paranoia of Gilead finds personification in Ofglen. We don't know whether she's a total cipher—a "trained pig" who does what her master(s) orders—or a secret rebel who cues our narrator in to an underground network of dissenters.
The narrator has only been at the Commander's house a short time, and she's only known Ofglen for part of that: "This woman has been my partner for two weeks. I don't know what happened to the one before" (4.18). We only see Ofglen in her context as a Handmaid, in her interactions with the narrator. Her life before the Republic is a blank. She wasn't at the Center with the narrator like many of the other Handmaids.
When the narrator first meets Ofglen, she says:
This one is a little plumper than I am. Her eyes are brown. Her name is Ofglen, and that's about all I know about her. She walks demurely, head down, red-gloved hands clasped in front, with short little steps like a trained pig's, on its hind legs. (2.19)
Wow, not exactly a kind description is it? The narrator emphasizes her similarity to this other Handmaid by referring to her as "this one," while also criticizing her as walking like a "trained pig." Perhaps she can't trust Ofglen because she suspects her of being a "true believer," or full participant in Gileadean society. Of the three things the narrator knows about Ofglen—her weight, eye color, and name/title—one of them (her name) is not even real.
Even as the narrator distrusts Ofglen, however, she realizes that deep down the two of them have no choice but to act the way they do:
I think of her as a woman for whom every act is done for show, is acting rather than a real act. She does such things to look good, I think. She's out to make the best of it [...] But that is what I must look like to her, as well. How can it be otherwise? (6.11-12)
The narrator seems to be criticizing Ofglen in part for the things she doesn't like about herself. Yet the narrator eventually realizes that Ofglen is a potential friend and ally, a member of the resistance. That is, if Ofglen is to be believed. (Can anyone in this world of Gilead ever be trusted?) The fact that Ofglen wants the narrator to pass on information about the Commander to her could either mean that she's part of the resistance or that she's a government spy who's trying to entrap her. On the other hand, the fact that she shares the resistance password, "Mayday," with the narrator would seem to indicate that she is a legitimate member of the resistance. What do you think?
Close to the end of the book, another woman replaces Ofglen, just as Ofglen had replaced her predecessor at the beginning:
"I am Ofglen," the woman says. Word perfect. And of course she is, the new one, and Ofglen, wherever she is, is no longer Ofglen. I never did know her real name. That is how you can get lost, in a sea of names. It wouldn't be easy to find her, now. (44.15)
All of the Handmaids at this posting (Glen's) are called Ofglen (of Glen). That's why the narrator is able to say of this third Ofglen, "of course she is" Ofglen. The name "Ofglen" says nothing about who this woman really is; her only identity is as the Commander's property. While the narrator despairs of "find[ing] her" friend here, that point seems to become moot just a few paragraphs later, when this new Ofglen reveals that the previous one hanged herself. The narrator feels relief that Ofglen may have died before giving up valuable information about her, then pauses to wonder whether that's even true.
The woman known as Ofglen for most of the book (which sounds kind of like The Artist Formerly Known as Prince) and the two doppelgangers that bookend her provide a contrast to the narrator; they are what she doesn't want to become. The narrator never learns Ofglen's real name, just as we never learn hers. Both Ofglens replace another one who hanged herself, just like the narrator.
What happens to Ofglen and her predecessor(s) is a warning to the narrator about what may happen to her. But that warning may come too late. Considering that the first Ofglen is a member of the resistance and had more knowledge about it than the narrator ever did, her death could mean the narrator is in even more danger than she realized.