O'Connor uses the image of mirroring and "the double" repeatedly during the story. How do they develop O'Connor's theme and her study of racism?
A good fun question! The doubling works in relation to the theme by making the bus, the story, and the society places full of mirrors (as you mention). This means that no one in the story can freely look down on someone else. If he or she does so, he or she is looking into a mirror. They share situations, characteristics, and therefore humanity. This doubling gives O'Connor a way to have her story's structure comment on racism without her having to do so directly. (Isn't she great!)
One more comment on mirroring: it is through this that her theme "everything rises must converge" develops. All of us, no matter how different we may seem, are part of a human community and as such share responsibility for its weaknesses, its (and I think O'Connor might use this word) "sins." In understanding this, we will rise and meet each other in a fuller, more just society. Indeed, even if we don't understand this, at some point we are confronted with evidence of this relationship, this burden we share.