What is the term for a sentence which reads same forwards and backwards?

Please note, I'm not asking for a palindrome. I mean to say that only the word order is rearranged, not the actual spelling of the word. An example might be as follows:

First ladies rule the State and state the rule: ladies first.

Solution 1:

They are still called palindromes, but are qualified by the term word-unit.

There are also word-unit palindromes in which the unit of reversal is the word ("Is it crazy how saying sentences backwards creates backwards sentences saying how crazy it is?"). Word-unit palindromes were made popular in the recreational linguistics community by J. A. Lindon in the 1960s. Occasional examples in English were created in the 19th century. Several in French and Latin date to the Middle Ages. - wikipedia

Solution 2:

Chiasmus, would be my response.

Taken from literarydevices.net:

Chiasmus is a rhetorical device in which two or more clauses are balanced against each other by the reversal of their structures in order to produce an artistic effect.
Let us try to understand chiasmus with the help of an example:

“Never let a Fool Kiss You or a Kiss Fool You.”

So in the example above, 'fool' and 'kiss' swap positions as noun and verb from the first usage to the second.

Incidentally, this is also a great example of zeugma.

In the OPs example, the first clause is mirrored in the second clause with the connective 'and' marking the mid point.