The Bog Song
Einar was born in a bog, and Lili is "a little girl born as a boy on the bog" (19.15). Maybe Einar/Lily's middle name is quagmire? Einar Quagmire Wegener has a nice ring to it, plus everyone seems to know he was born in a bog: "Born in a bog" (2.12) is a shorthand way his students describe his personality. Because apparently that explains everything.
In his pre-Lili days, Einar paints the bog and sings a song about a bog: "There once was an old man who lived on a bog and his pretty little son, and their lazy little dog" (3.22). It's a pretty boring song in our humble opinion, but more importantly, it's clearly about Einar, his father, and their little dog Edvard II. And that Einar describes himself as "pretty" matters—as the story unfolds, we watch him increasingly identify as female, and pretty is a term often reserved for women and feminine things.
Thing is, Einar's life doesn't just start in the bog; it pretty much is a bog. It's all muddles and murky, particularly when it comes to his gender identity. Fittingly, when Einar decides he wants to just be Lili, he stops painting the bog: "'I'm having a hard time imagining the bog,' he'd call from his studio, where his canvases and his paints were kept tidy" (14.4). When Einar becomes Lili, his internal murkiness subsides and the waters become clear.