I think The Crying Game switches genres enough and the genre-switching is so central to the entire movie that it can't be pinned down to any genre by the end. I'd say it's about identity - how it's defined, how it's chosen, and what aspects of it people choose to use to define themselves. But that's a theme, not a genre. Looking back from the end, though, my gut reaction is that it's a love story. Especially since the plot is a variant of a classic love story in which a soldier who goes off to war, leaving his girlfriend behind, tells his buddy to look her up if he doesn't make it, and then the buddy goes home and falls in love with her himself.

I know there's more anime than Madoka that have mid-story shifts, but a lot of them are primarily tonal rather than genre per se. Then again, you could argue that the shift in Madoka is also tonal. That being said... Neon Genesis Evangelion, Trigun, Escaflowne... Princess Tutu doesn't exactly shift genres, but the second half is radically different from the first. Utena, same.

In books, Stephen King's Dark Tower series has a minimum of four different genre switches in a seven-book series. Rosemary Kirstein's Steerswoman has a big genre switch somewhat early on, and then a sub genre switch later, with plenty of time to explore the implications of both.

But my absolute best example of a mid-book genre switch is Frances Hardinge's Cuckoo Song. It starts off as horror and becomes a different genre entirely about half or a third of the way in. I was not expecting that at all.

Also, you never know what genre any given Diana Wynne Jones book will end up in. Like, I'm not sure how to classify The Homeward Bounders to begin with, but whatever it is when it starts out isn't where it is by the two-thirds point, let alone where it is at the end. Hexwood also switches genres at least twice during the story.