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What are statements where same two words that appear initially are interchanged later to give a completely different meaning called? Eg: Eat to live, don't live to eat. Eg2 : don't love the ones who hurt you, and don't hurt the ones who love you.

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I'm pretty sure these are cases of "antimetabole" that you're referring to:

Antimetabole...

is the repetition of words in successive clauses, but in transposed order; for example, "I know what I like, and I like what I know".
Antimetabole Wikipedia article

Some examples from the article:

  • "Eat to live, not live to eat." Attributed to Socrates (also the example in your question)
  • "If you can't be with the one you love, love the one you're with."
  • "Failing to prepare is preparing to fail."

There are many more in the link I provided.

This is a very easy thing to do. In the 1999 comedy movie Mystery Men such ways of speaking is lampooned in one particular character named the Sphynx, the superheroes' trainer. One such instance is supposed be to humorous.

"He who questions training only trains himself at asking questions."

link to quote in video

He is then derided for speaking in such aphoristic ways.

Wikipedia describes the Sphynx as speaking in "chiasmus". It is noted in the chiasmus article that antimetabole is a subtype of "chiasmus".
Chiasmus Wikipedia article

Another online source explaining this and with examples is at literarydevices.net

Of note on this site is this line:

Antimetabole and chiasmus are very closely related, and some experts even use them interchangeably.

You can read into the exact distinctions if you're interested.

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Here are some other interesting sentences like the ones you have quoted. All these sentences are given here as examples of the figure of speech epigram.

  • “Mankind must put an end to war, or war will put and end to mankind.” – John F. Kennedy
  • “If we don’t end war, war will end us.” – G. Wells
  • “Live simply, so that others may simply live.” – Mother Teresa

But the term epigram is not restricted or limited only to statements as you stated.

"Epigram is a rhetorical device that is a memorable, brief, interesting, and surprising satirical statement".(literarydevices.net)

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