Both Alice's Adventures in Wonderland and Through the Looking Glass are books about weird, magical journeys when things that seem mundane—playing cards, rabbits—turn out to be extraordinary. Aside from the talking rabbits, it sounds a lot like The View from Saturday.
We could call these two books intertexts—that's a fancy way of saying that The View from Saturday is having a kind of literary chat with Lewis Carroll. But we could also just say that the books symbolize the journeys that all of The Souls are on. Mr. Singh lays it out for Mrs. Olinski, and, conveniently, us:
Think about the question that Maxwell missed. They did not know that Humpty Dumpty was not in Wonderland because they never journeyed through the looking glass to find him. How can you know what is missing if you've never met it? You must know of something's existence before you can notice its absence. So it was with The Souls. They found on their journeys what you found at Sillington House. (11.17, 27)
In other words, Maxwell can't win the competition because they've never been on a journey to find the missing piece. They've never been picked up by the shell and plopped down in an unfamiliar location. They've never had to balance, because they've never been missing a limb.
And one more analogy to hit all the symbols: They've never had an epiphany, because they've never known they didn't know something.