First, as an example of what follows, here is a recapitulation of the chiasm in Hebrews 12:1-2 posted a few days ago:

  

having SEATED around us such a great cloud of witnesses

       

     SETTING ASIDE every weight and entangling sin

          with ENDURANCE

               let us run the race THAT IS SET BEFORE US

                    fixing our eyes on the founder and
                    perfecter of our faith, Jesus

 

               who for the joy THAT WAS SET BEFORE HIM

          ENDURED the cross

     DISREGARDING the disgrace,

and HAS TAKEN HIS SEAT at the right hand of the throne of God.

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Second, a bit of discussion on chiasmus:

A literary chiasm occurs when elements of a given literary unit are structured and balanced symmetrically in an inverted parallelism. A brief example is found in 1 John 3:9: “(A) Whoever has been born of God (B) does not sin (C) because his seed remains in him (B’) and he is not able to sin (A’) because he has been born of God.”

The chiastic form is still commonly used, though not with near the ubiquity it seems to have enjoyed in antiquity. Although the use of chiasmus in Scripture was noted explicitly as early as the mid-1800’s, the form did not receive significant attention until the seminal works of Nils Lund two centuries later (a series of articles in the 1930’s culminated in his Chiasmus in the New Testament [University of North Carolina Press, 1942]). Since Lund, a constant stream of works has sought to define and delimit chiasmus in Scripture and other ancient literature. Within both Old and New Testaments, chiasmus has been proposed for literary units ranging from single verses to entire books. At one end of the scale, concise chiasms with only two or three sets of parallel elements are routinely acknowledged; conversely, the existence of intentional chiastic structures of extensive length has been strongly debated.

Proposals of chiastic structure in Scripture have at times reflected more scholarly ingenuity than authorial intent! The more Procrustean applications of the chiastic form to Scripture, however, have brought about increasingly sophisticated attempts to develop criteria for identifying chiasmus.

There are two major exegetical benefits to recognizing the presence of chiasmus in a given passage.  First, the central element is generally understood to be the strongest point of emphasis in the chiastic text, as in Hebrews 12:1-2.  Second, a better understanding of a particular word or phrase may be gained by comparing it to its chiastic counterpart; an element in the second half of a chiasm frequently intensifies, specifies, or completes its parallel counterpart.

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Third, resources for further study:

For those interested in this phenomenon, there is a website devoted to its use in Scripture.  Also available is a bibliography of works involving chiasmus.  The following is a list of some of the better articles on chiasmus, and a few major works on the subject (Parunak’s article is particularly good, and Thomson’s work is excellent and balanced):

Aune, David E. “Chiasmus.” Pages 93-96 in The Westminster Dictionary of New Testament and Early Christian Literature and Rhetoric. Louisville: Westminster John Knox, 2003.

Breck, John. The Shape of Biblical Language: Chiasmus in the Scriptures and Beyond. Crestwood, N.Y.: St. Vladimir’s Seminary, 1994.

Brouwer, Wayne. The Literary Development of John 13-16: A Chiastic Reading. Society of Biblical Literature Dissertation Series 182. Atlanta: Society of Biblical Literature, 2000.

Man, Ronald E. “The Value of Chiasm for New Testament Interpretation.” Bibliotheca Sacra 141 (1984): 146-157.

McCoy, Brad. “Chiasmus: An Important Structural Device Commonly Found in Biblical Literature.” Chafer Theological Journal 9:2 (2003): 18-34.

Parunak, H. Van Dyke. “Oral Typesetting: Some Uses of Biblical Structure.” Biblica 62 (1981): 153-68.

Porter, Stanley E. and Jeffrey T. Reed. “Philippians as a Macro-Chiasm and Its Exegetical Significance.” New Testament Studies 44 (1998): 213-31.

Stock, Augustine. “Chiastic Awareness and Education in Antiquity.” Biblical Theology Bulletin 14 (1984): 23-27.

Thomson, Ian H. Chiasmus in the Pauline Letters. Sheffield: Sheffield Academic Press, 1995.

Welch, John W., ed. Chiasmus in Antiquity: Structures, Analyses, Exegesis. Hildesheim: Gerstenberg Verlag, 1981.

Welch, John W. “Criteria for Identifying the Presence of Chiasmus.” Journal of Book of Mormon Studies 4:2 (1995): 1-14.