grammaticality - Why do we use the word "Do" when connecting a sentence?

I was reading a news paper article of Times Of India, and came across a sentence-

To begin with, a woman's right to property has already been established under law. This means that she has equal rights to her parental property as her male siblings. In such a scenario, according women an extra legal right over their husbands' residential property - which too could be inherited - is unfair. Neither do men have the same rights over their wives' property nor can they claim emotional damages during divorce.

Why there is a word "do" in between "Neither" and "men have".?

From my pointing of view It might be -
"Neither men have the same rights over their wives' property nor can they claim emotional dames during divorce".

I have also heard people saying that
"I do agree with your statement".

Why could not it may be "I agree with your statement".

Is There any grammatical mistake in these sentences or both I can use interchangeably?

Answer

The most common form of sentence inversion in English is called subject-auxiliary inversion. In order to perform this inversion, the sentence needs an auxiliary verb. If the sentence doesn't have an auxiliary, one is added. Often, that auxiliary is "do".

Inversions are most commonly used for questions. In this case, the inversion is used to create an embedded question, and there are actually two embedded questions in the sentence each with inversions. The first one uses "do" and the second one uses "can".

Neither

• do men have the same rights over their wives' property

nor

• can they claim emotional damages during divorce.

The embedded questions are complete sentences in this case, so you can say:

• Do men have the same rights over their wives' property?


• Can they claim emotional damages during divorce?

Note that not all embedded questions use question order, e.g.,

I know who he is. (correct)

I know who is he. (incorrect)

The second example is different.

I do agree with your statement. (correct)

I agree with your statement. (correct)

In this case, "do" is used for emphasis.