Why are the subject and the verb inverted after the conjunction 'as'?

Question

These are from the page for 'as' as conjunction on Oxford Dictionary. I don't understand why they are inverted like 'as is her mother' or 'as do I'. Why not 'as her mother is' or 'as I do'?

I've searched the net a little and people are saying that 'as do I' is more formal than 'so do I', but is that really so? I've learned 'so do I' to mean 'me too' long time ago in the official text book written by the government in my country, but I don't think I ever learned 'as do I'. I've just studied that 'as' can be a relative pronoun but replacing 'which' to 'as' doesn't make sense for these two sentences, nor giving 'it' after 'as', so I know they are not relative pronoun, but what are they?

4 used to make a comment or to add information about what you have just said

• As you know, Julia is leaving soon.

• She's very tall, as is her mother.

As is a conjunction and an adverb and is used before a clause, another adverb, or a phrase beginning with a preposition: She enjoys all kinds of music, as do I. As always, he said little.

Answer 1

The word as in your examples is a conjunction.

It is used as a conjunction to express similarity. You can think of it (this as) as like, where like can be used as a more informal version of as. You can write it in normal word order ("as her mother is" or "as I do"), but keep in mind that you could also find the inverted word order used in a very formal style.

Here are some related sub-entries from Practical English Usage by Michael Swan.

326 like and as: similarity, function

We can use like or as to say that things are similar. We can also use as to talk about function – the jobs that people or things do.

326.2 as (similarity): as I do
As is a conjunction. We use it before a clause, and before an expression beginning with a preposition.
　　　 as + clause
　　　 as + preposition phrase
　　　 Nobody knows her as I do.
　　　 We often drink tea with the meal, as they do in China.
　　　 [...]
326.4 inverted word order: as did all his family
In a very formal style, as is sometimes followed by auxiliary verb + subject (note the inverted word order – see 302).
　　　 She was a Catholic, as were most of her friends.
　　　 He believed, as did all his family, that the king was their supreme lord.

Here is the related part of the entry 302 mentioned above.

302 inversion (1): auxiliary verb before subject

302.4 after so, neither, nor
In 'short answers' and similar structures, these words are followed by auxiliary verb + subject.
　　　 I'm hungry. ~ So am I.
　　　 I don't like opera. ~ Neither/Nor do I.

302.5 after as, than and so
Inversion sometimes happens after as, than and so in a literary style.
　　　 She was very religious, as were most of her friends.
　　　 City dwellers have a higher death rate than do country people.
　　　 So ridiculous did she look that everybody burst out laughing.

Answer 2

I don't understand why they are inverted like 'as is her mother' or 'as do I'. Why not 'as her mother is' or 'as I do'?

Both are acceptable. In American English, "as do I" is the more formal and precise of the two locutions. A well-educated American might say it either way; but one would not expect to hear "as do I" from an American who had never finished high-school, for example.