How does Lewis Carroll play with language in Through the Looking-Glass?

In Through the Looking-Glass, Alice encounters a variety of bizarre situations, many of which are linguistic in nature. While we often use the term “play with language” somewhat loosely, Carroll’s novel often quite literally plays with language to produce games and puzzles for its reader.
Consider “Jabberwocky,” the often discussed poem within the novel. When Alice first happens upon the book that contains it, she decides that she cannot read it “for it’s all in some language I don’t know”:

She puzzled over this for some time, but at last a bright thought struck her. “Why, it’s a looking-glass book, of course! And if I hold it up to a glass, the words will all go the right way again.”

The passage is printed backwards, and if the reader holds it up to a mirror, it appears legibly. This moment of linguistic play produces what Alice refers to as a puzzle—a game for the reader to play.
But there is a joke in Alice’s initial reaction that has to do with the linguistic play present throughout “Jabberwocky.” Once we read “Jabberwocky,” we see that there are words that we do not know:

‘Twas brillig, and the slithy toves
Did gyre and gimble in the wabe.

But as we read through the poem, we can make sense of it. As readers we can interpret the passage because of the syntactical relationship that exists between words, even when we do not know what these words mean. Imagining their meaning becomes another sort of game precisely because the words that we play with are made up. This dynamic is in play throughout the novel: Alice must constantly manipulate language to solve the puzzles that other characters present to her. Characters use a precision of language that often twists the meaning of words entirely—as if they are holding language up to a mirror.