"Inversion" is actually a rhetoric device and it means, changing the usual [word order] to achieve a certain effect. So we are, in its origin, talking about a grammatical phenomenon.
1a. "What a lovely day", I thought, and I turned around to watch the flowers. (normal)
2a. "What a lovely day", thought I and around I turned, to watch the flowers. (inversion)
In satire, inversion is used as a rhetorical device on a broader scale: there, it is turning "the usual" [situation] around, usually into its opposite.
Example (and a book I have read): the story about a country in which sex in public is widely accepted, people have it and talk about it all the time - but food is an absolute taboo topic, people do not eat in public because it is thought to be inappropriate.
Here, "the usual" situation (food being "okay", sex being taboo in public) is "turned around", to make fun of the actual situation, i.e. critizing the rather prude attitude towards sex in the US, for instance.
Hope that helped.