Symbolism, Imagery, Allegory
When you're just reading Cloud 9, it's easy to lose track of who's playing which character. But watching—or even imagining—the play gives you a sense of just how disorienting Churchill's casting specifications are.
Quick recap: Betty (in Act 1) is played by a man. Edward (in Act 1) is played by a woman. Joshua, the black servant, is played by a white man. Cathy is played by a man. For a peek at how the play might look on stage, just check out a few of these clips: Clip 1, Clip 2, Clip 3.
Churchill's point, and the crux of this (pretty heavy) symbolism, is the absurdity of gender or racial roles. She's also pointing out how these roles' absurdity often goes unquestioned.
It might seem absurd for a woman to say, "The whole aim of my life/ Is to be what he [Clive] looks for in a wife" (1.1.18-19), but it's extra absurd for a man dressed as a woman to be saying that. The audience is forced to think about the constraints of traditional feminine roles by having female lines spoken by a man in drag.
When we watch Joshua say, "My skin is black but oh my soul is white" (1.1.24), the audience does a double-take because the actor playing Joshua does have white skin. The inherent insanity of racism is underlined through Churchill's casting specifications.
Is it confusing? Yes. Is it thought-provoking? Oh hecky yeah. It's also hilarious. And that's what Churchill is doing here. Her casting choices make the audience simultaneously uneasy and amused. And uneasy amusement is what Cloud 9 is all about, at its core.