Q:

"We must eat to live, not live to eat." — What kind of rhetorical figure is that?

In this kind of sentence the second half is an inversion of the first half. Is it a rhetorical artífice? What kind?

A:

This is a type of chiasmus (in general) or antimetabole (to be specific).
Example of antimetabole from Sylvae Rhetoricae:

You can take the gorilla out of the jungle, but you can't take the jungle out of the gorilla.

