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On Chiasmus and Antimetabole
All our sources agree that while antimetabole is an inversion of repeated words in successive clauses, chiasmus is an inversion of parallel grammatical structures only. Nevertheless, in critical practice it has become so common to use chiasmus for both types that many expert critics will understand you when you say chiasmus and not have the faintest idea what antimetabole is. The easiest way to understand it is to think of antimetabole as a special sub-class of chiasmus. Rarely, however, do you need to make such a distinction. When in doubt, use chiasmus.

Basic Definitions
Chiasmus in Handlist of Rhetorical Devices
Chiasmus in Silva Rhetoricae.
Antimetabole in Silva Rhetoricae.

Corbett and Connors 394-395: Antimetabole: Repetition of words, in successive clauses, in reverse grammatical order. Chiasmus: Reversal of grammatical structures in successive phrases or clauses.
Notes on Chiasmus
Chiasmus is also related to a tactic featured under arrangement known as the inclusio or envelope pattern, whereby a text exhibits repetitive structures moving outward from a central point, often notated
ABC-C'B'A' and so forth.

Illustrations

"Ask not what your country can do for you; ask what you can do for your country." John F. Kennedy, Inaugural.
 
References
Corbett, Edward P. J. and Robert J. Connors. Classical Rhetoric for the Modern Student. 4th ed. New York: Oxford UP, 1999.
Davis, William L. "Better a Witty Fool than a Foolish Wit: The Art of Shakespeare's Chiasmus." Text and Performance Quarterly 23 (2003): 311-330.
Gardner, A. Edward. "The Return of the Beloved: The Chiasmus and the Messianic Secret of Abraham Lincoln." Central States Speech Journal 38 (1987): 133–151.
Grothe, Mardy. Chiasmus.com. http://www.chiasmus.com/
---. Never Let a Fool Kiss You or a Kiss Fool You: Chiasmus and a World of Quotations That Say What They Mean and Mean What They Say. New York: Viking Penguin, 1999.
Lynch, Christopher. "Reaffirmation of God's Anointed Prophet: The Use of Chiasm in Martin Luther King's "Mountaintop" Speech." Howard Journal of Communication 6 (1995): 12-31.