Why do you say "so do I"?

Why is the order of the words in "so do I" or "nor do I" different from the normal order?

With respect to information structure, this word order pattern seems to be equal to fronted adverbials, as in:

Into the room walked a man

The new information / focus follows the finite verb. Similarly, in "So do I", "so" refers to the proposition in the previous utterance, "do" is a dummy verb, and "I", the new information, comes last.

To agree with a positive statement:

We use so + auxiliary/modal verb + pronoun:

"I like tea without sugar.'
'So do I.'

To agree with a negative statement:

We use nor/neither + auxiliary/modal verb + pronoun:

"I don't like tea with sugar.'
'Nor do I.' or 'Neither do I.'

To disagree with a positive statement:

We use pronoun + auxiliary/modal verb + not (-n't):

"I like tea without sugar.'
'I don't.'

To disagree with a negative statement:

We use pronoun + auxiliary/modal verb:

"I don't like tea with sugar.'
'I do.'

@itrekkie: I think you're on the right track.

Note that 'so' here has a different meaning from that it would have in the normal order ("I do so") - it means 'also', or 'as well' and it seems to me that it can have that meaning only when fronted. The fronting is clearly for emphasis, but that in itself is not enough to explain why the fronting is obligatory for that meaning. "As do I" and "Nor do I" don't have "as" or "nor" displaced, but they still have the inversion, and it seems to be crucial to this "as well" meaning. You can say "As I do", but without that meaning; and for some people "Nor I do" exists, but it means "and I do not", rather than "I do not either".

"Do" is syntactically a full verb, but semantically it's a place holder. But it's not crucial here: "So have I" and "So am I" are normal.

This isn't related to the issue of fronted "adverbials" as @Arne's accepted answer suggests. Those "adverbials" are usually prepositional phrases, as in the example given, which come after intransitive verbs. They don't require an auxiliary verb, and they don't work with transitive verbs. Consider:

• the plane flew over the fence --> over the fence flew the plane
• the boy flew the plane over the fence --> *over the fence flew the plane the boy

You can, however, do the "nor flip" with transitive verbs, and it does require an auxiliary verb, for example:

• nor will you see him again

You can do the same thing with a number of negative and limiting adverbs, and I think this is the pattern that nor belongs to.

• never did I say such a thing
• rarely is it on time
• seldom can they be trusted

Although it is mostly like a conjunction now, nor began life as an adverb according to the OED. The first example with so doesn't fit this pattern well because so is neither negative nor limiting. Nevertheless, this is the closest fit I can see.