Symbolism, Imagery, Allegory
The first time mirrors really come up is in the treatise on the Steppenwolf (the strange book that Harry is given that is all about himself). It says that:
It is possible that Harry will one day be led to [humor]. It is possible that he will learn one day to know himself. He may get hold of one of our little mirrors. (52)
It seems like the book is saying that Harry has something he needs to learn, and that it's going to have something to do with self-knowledge. In fact, the little mirror comes in pretty handy when Harry finally makes it to the Magic Theater. Pablo pulls a mirror out of his pocket, and tells Harry to use it to see himself:
He held the little glass before my eyes […] and I saw, though indistinctly and cloudily, the reflection of an uneasy, self-tormented, inwardly laboring and seething being—myself, Harry Haller. And within him again I saw the Steppenwolf, a shy beautiful dazed wolf with frightened eyes that smoldered now and again with anger, now with sadness. (495)
Pablo's mirror helps Harry to see how he has been considering himself all of these years, as a two-souled being in constant battle with himself. The mirror will also come in handy for helping Harry to banish that boring, binary image of himself and see more options:
The mournful image in the glass gave a final convulsion and vanished. The glass itself turned gray and charred and opaque, as though it had been burned. With a laugh Pablo threw the thing away and it went rolling down the endless corridor and disappeared.
"Well laughed, Harry," cried Pablo. "[…] You have done with the Steppenwolf at last. It's no good with a razor. Take care that he stays dead." (506-507)
By getting rid of the figures in the mirror, Harry is kind of like a clean slate, all new for whatever the magic theater has to teach him, and for whatever new selves he wants to try on.
Later, a much bigger mirror will come into play, and offers Harry many selves to choose from:
But I scarcely had time to recognize myself before the reflection fell to pieces. A second, a third, a tenth, a twentieth figure sprang from it till the whole gigantic mirror was full of nothing but Harrys or bits of him, each of which I saw only for the instant of recognition. (509)
Now the mirror is gigantic, not a little one, which gives Harry room to imagine. He can be any of his selves that he chooses, young or old. This means that his life is not locked down into either man or wolf; rather, he has tons of adventures to live through. Woo-hoo!
A mirror will be used once again to break Harry's personality into pieces when he learns how to play chess with himself:
He held a glass up to me and again I saw the unity of my personality broken up into many selves whose number seemed even to have increased. (592)
Are you starting to see a pattern? Every time Harry looks into a mirror, his personalities multiply, breaking up so that he is more and more free from being the boring ol' Steppenwolf.
Harry starts to ruin the theater, though, when he runs away from the wolf-man scene. He finds the mirror again, but instead of seeing thousands of himself, he only sees one:
There was no wolf in the mirror, lolling his tongue in his maw. It was I, Harry. My face was gray, forsaken of all fancies, wearied by all vice, horribly pale. […]
"Harry," I said, "what are you doing there?"
"Nothing," said he in the mirror, "I am only waiting. I am waiting for death." (619-621)
This isn't what Harry wants to hear, and he'll take it out on the boring, old, gray Harry in the mirror later on. He is so frustrated by the problems in life that he forgets all about his mission to learn to laugh and turns suicidal again:
Bah, the devil—how bitter the taste of life! I spat at Harry in the looking glass. I gave him a kick and kicked him to splinters. (650)
In one action Harry tries to destroy himself, but he only destroys his reflection. Just like all the times that he thinks about or tries suicide, he can't actually do away with himself… because he isn't just one soul or two. As Pablo tries to teach him, he contains an endless number of individuals.