Chiasmus is a stylistic figure in literature or rhetoric in which a pair (or more) of words (or ideas) in the first part of a sentence are reversed in the second part, to great effect. It demonstrates a ‘crossover’, which you could represent diagrammatically by an X. The Greek letter chi is x-shaped – hence ‘chiasmus’.

One of the most famous modern ones is JFK's "...ask not what your country can do for you; ask what you can do for your country."
Kennedy used chiasmus frequently in his speeches, such as "Let us never fear to negotiate, but let us never negotiate out of fear" and "mankind must put an end to war or war will put an end to mankind."

He wasn’t the only political orator who saw the usefulness of this construction. Bro Bob Dole said, "A government that seizes control of the economy for the good of the people ends up seizing control of the people for the good of the economy."

One good example of a three-part chiasmus is well known to Freemasons:
A  Whoever sheds
B  the blood
C  of man
C  by man
B  shall his blood
A  be shed
Genesis 9:6

Other chiastic sayings:

• When you are young, everyone else looks old; when you are old, everyone else looks young.

• As you immerse yourself in thinking about them, they begin to immerse themselves in your thinking.

• Queen Elizabeth said to Sir Walter Raleigh, who brought tobacco back to England from the New World: I have seen men turn gold into smoke, but you are the first man to turn smoke into gold.

• For my thoughts are not your thoughts, Neither are your ways my ways, saith the Lord. (Isaiah 55:8)

Chiasmus was indeed favored by Hebrew writers, and can get more complex than just a simple crossover of words. Chiasmus can express ideas in a repetitive rising and falling crossover pattern:

A For this people's heart is become gross
B and their ears are dull of hearing,
C and their eyes they have closed
C lest at any time they should see with their eyes
B and hear with their ears
A and should understand with their heart

Matthew 13:15

A chiasm of five elements:

        A  Save me
          B   O my God,
            C  For thou has smitten
              D  All my enemies
                E   On the cheek-bone
                E   The teeth
              D   Of the wicked
            C   Thou has broken.
         B   To Yahweh
        A  The salvation.
    

Psalm 3:7-8

And of seven elements:

A  Arise,
       B  Shine,
           C  For thy light is come,
              D  And the glory
                E  Of Yahweh
                 F  Upon thee is risen
                    G  For behold, dimness shall cover the earth
                    G   And gross darkness the peoples.
                  F  But upon thee will arise
                E  Yahweh
              D  And his glory shall upon thee be seen
            C  And nations shall come to thy light
      B  And kings to the brightness
A Of thy rising.

Isaiah 60:1-3

- this is the text from which we get the phrase ‘rise and shine’.

There’s often much more that meets the eye in the writing of the scriptures. Nonetheless, the straightforward crossovers can be more memorable (and occasionally fun):

• America did not invent human rights. It is the other way round. Human rights invented America.

-Jimmy Carter, farewell address.

• It's not the men in your life that counts, it's the life in your men."
  Mae West in I'm No Angel (1933)

• I’d rather have a bottle in front of me than a frontal lobotomy.

To add to the fun, chiasmus can actually be implied when the original unreversed phrase is well known:

• A hard man is good to find. - Mae West
• In the beginning Man created God.  - “Jethro Tull”
• Time's fun when you're having flies. - “Kermit the Frog”