Although his most famous line is a chiasmus, I didn't realize JFK was a compulsive user of the form. For example:
It is not our wealth that built our roads, but it is our roads that built our wealth
Do not pray for tasks equal to your powers. Pray for powers equal to your tasks.
Liberty without learning is always in peril; learning without liberty is always in vain.
Man must put an end to war or war will put an end to mankind.
The weapons of war must be abolished before they abolish us. [arguably]
And so, my fellow Americans: ask not what your country can do for you — ask what you can do for your country.
Let us never negotiate out of fear. But let us never fear to negotiate.
Something else that he appears to have been fond of, as per his Wikiquote page, is grotesque internal rhymes.
Not the peace of the grave or the security of the slave.
Our goal is not the victory of might but the vindication of right. [followed, one sentence later, by "thank you and good night"]
Terror is not a new weapon. Throughout history it has been used by those who could not prevail, either by persuasion or example. But inevitably they fail.
I wonder if he (or the relevant speechwriters) ever got to the stage where they reflexively checked every sentence to see if it made sense transposed.