Is there a name for inverting word order to accomplish a different meaning?

Antimetabole is the word for this. Quoting from Wheeler,

ANTIMETABOLE (Greek, "turning about"): A rhetorical scheme involving repetition in reverse order: "One should eat to live, not live to eat." Or, "You like it; it likes you." The witches in that Scottish play chant, "Fair is foul and foul is fair." One character in Love's Labor's Lost uses antimetabole when he asks "I pretty, and my saying apt? Or I apt, and my saying pretty?" (I, ii). Antimetabole often overlaps with chiasmus. This device is also called epanados. See schemes.

Note, while chiasmus is nicely related to antimetabole, epanados really is no more antimetabole than are any of antanaclasis, "the stylistic trope of repeating a single word, but with a different meaning each time", or ploce, "a figure of speech in which a word is separated or repeated by way of emphasis; the repetition of a word functioning as a different part of speech or in different contexts" or conduplicatio, "the repetition of a word in various places throughout a paragraph". Antimetabole is slightly related to anadiplosis, "the repetition of the last word of a preceding clause" via the y repetition in the x y y x pattern of antimetabole.