So do I, etc

These expressions are quite informal, Martine, and are readily used in short answers in spoken English to express agreement with what has been said in the first statement. So is here followed by inverted word order: auxiliary verb + subject:

• Judy can run 100 m in 11 seconds. ~ So can Henry!
• I've got a blister on my big toe. ~ So have I.
• I'm going to get it seen to by the club doctor. ~ So am I.
• I like to eat really hot food on cold days and so do all my friends.

So is occasionally followed by normal word order in short answers to express surprised agreement:

• If you don't believe me, just look out of the window. It's snowing! ~ So it is!
• You've given me tea and I asked for coffee! ~ So I have! I'm sorry.

Neither / nor would I, etc

These expressions are used in a similar way to So would I, etc, to express agreement with negative statements:

• I can't swim very well and neither can my sister. 
• I wouldn't dream of going into the water if the temperature is below 20° C and nor would any southerner.

inversion after negative expressions

We can use inversion in statements for the purpose of emphasis if we decide to start the statement with a negative expression. Compare the following:

• Under no circumstances would I wear a mini-skirt.
• I wouldn't wear a mini-skirt under any circumstances.

In this example, the first statement is more emphatic than the second one. We can use this approach with a wide variety of adverbial negative expressions, although it makes them sound rather formal. Compare the following:

• At no time would he allow his team mates to argue with the referee.
• Rarely / Seldom have I seen such an exciting game of football.
• Hardly had I taken my seat before two goals were scored.
• I had to show him my press pass and only then did he let me in.
• Only when the players had changed into smart clothes after the match were they allowed to talk to the TV reporters.

Inversion in conditional sentences

We can use inversion in certain types of conditional sentences when the if-clause begins with had, were or should. Sentences with inversion sometimes sound more formal than those with the more conventional if-construction. Compare the following:

• Had he not resigned, we would have been obliged to give him the sack.
• If he had not resigned, we would have been forced to sack him.
• Were she to find out that he was seeing some one else, she'd go berserk.
• If she were to find out that he was cheating on her, she would go mad.
• Should you decide to cancel the contract, please let me know by Friday.
• If you decide to withdraw from the agreement, please phone me by Friday.