What do we 'turn round and say'?

As an American, I can't say I've heard this very often, either in person or in movies. I think it's more likely used in the UK. But I instantly recognize the meaning. It means to abruptly change tone of speech and/or behavior and confront another person in a critical, hostile or recalcitrant manner. Call it an idiom. In 'what if she turns round and refuses to pay?' she may abruptly change her prior promise to pay and decide to withhold payment. The main idea is the abrupt change of position.

I think it is just a pleonastic way to express the concept of something that may be said, to give more emphasis to the sentence!

You say:

I think the turning around is not about a reversal of a previous position, but is used to indicate that the remark came suddenly and out of nowhere. In other words the fact of turning around indicates that it is not part of an existing dialogue which had been going on.

Speaking as a Brit, I can confirm your summary to be completely accurate.