"Nice to see you, to see you nice!"

Look at this famous phrase used by a British talkshow host when saying goodbye to his audience:

'Nice to see you, to see you nice!'

Nobody in the UK (including my grandmother who was a frequent viewer) seemed to think it sounded strange...

I think it's grammatically acceptable - due to some missing but understood elements of the sentence.

Can you have a go at estimating what they may be?

Solution 1:

I'm not a native speaker, so I may get this completely wrong, but I would interpret it as:

I am happy to see you, and too see that you are well.

So, the second part would just be a reinforcement of the first one. In particular the repetition of "to see you" is used as a way to strenghten the premise, and probably because

Nice to see you nice!

is probably of less effect.

Solution 2:

This is an example of chiasmus, a rhetorical figure in which the structure of the first clause is reversed in the second clause, in an AB-BA pattern. One of the most famous examples comes from John F. Kennedy's presidential inauguration speech:

Ask not what your country can do for you; ask what you can do for your country.

This device is powerful, as are most rhetorical devices, and one should take care in using them. Idle usage can sound silly or pompous. As Ward Farnsworth notes in his exemplary Classical English Rhetoric:

A chiasmus that reverses the same words ... calls attention to itself strongly, and so must be used with particular care ... In the hands of a typical modern politician, [it] will sound disagreeably slick and perhaps even repulsive.

There is nothing ungrammatical about the usage in your example, but it sounds odd here because its power is way out of proportion to the simple and somewhat insipid greeting it's used to express.

Solution 3:

It really is just a catch phrase. The audience shouts out the second "nice" on cue - the second "nice to see you" is the cue for the audience to all shout out "nice" at the same time.