You are absolutely correct. "How do you do" is an old fashioned introduction and is an obsolete synonym of "hello", and consequently the proper response is "how do you do?".
Mr Darlington: How do you do, Mrs Windermere?
Mrs Windermere: How do you do, Mr Darlington?
This exchange is exactly equivalent to the more modern (and at the time, more vulgar):
Mr Darlington: Hello, Mrs Windermere.
Mrs Windermere: Hello, Mr Darlington.
Note that the phrase "how do you do" has long since gone out of fashion in British English, and was never in fashion in American English.
Generally speaking, native speakers in both British English and American English now use the phrase hello instead, although how do you do does still persist in some formal settings in British English, generally during formal introductions, and lives on as howdy in some dialects of American English (and like how do you do, the correct response to howdy is howdy).
Don't be surprised if using this phrase that you receive odd looks - the phrase is excessively polite, and may easily be mistaken (even by native speakers) for the more common introduction "How are you?", to which the proper response is "I am fine, thank you, and you?".
