Do you really answer "How do you do?" with "How do you do?"

As Cerberus wrote about 'U' English, replying to "How do you do" with "How do you do" used to be the case among some classes in England (at least), but it seems to be (sadly) nearly extinct. Kate Fox writes in the first chapter of Watching the English (which is about talking about the weather):

We used to have another option, at least for some social situations, but the ‘How do you do?’ greeting (to which the apparently ludicrous correct response is to repeat the question back ‘How do you do?’) is now regarded by many as somewhat archaic, and is no longer the universal standard greeting. The ‘Nice day, isn’t it?’ exchange must, however, be understood in the same light, and not taken literally: ‘How do you do?’ is not a real question about health or well-being, and ‘Nice day, isn’t it?’ is not a real question about the weather.

So in this usage, "How do you do?" wasn't an actual question about the person: it was just a meaningless greeting, and for instance what one might say when introduced to someone (in lieu of "Pleased to meet you"), presumably while tipping one's hat. In reply, the other person, also wishing to make the same meaningless greeting, would say "How do you do". Note that "How do you do" in this usage was even spoken as a statement, not as a question (i.e., without a rising tone at the end).

These days "How do you do?" is more likely to be interpreted as an actual question. Interestingly, as n0nChun observes, one does sometimes hear a similar exchange these days, with "What's up" getting the reply "What's up", or even just "sup" — "sup".

The reason for the varied responses is that the question you ask is not only about the English language, but it also touches on etiquette, which is more idiomatic.

The short answer is: yes, the answer to "How do you do?" is "How do you do?" It is also acceptable to reply "I'm fine, and you?" and similar variations.

However -- and this is important to the etiquette side of things and not so much the language side -- "how do you do?" is actually not a question! It is a greeting, similar to "good morning" and, hence, an appropriate reply is also "good morning to you!"

Etiquette allows simple replies such as "I'm fine, how are you?" to appease the literalists who might feel off-put by a non-reply to what they mis-interpret as having asked you a question. However, good manners also prevents one from discussing their personal troubles with every acquaintance or stranger on the street who greets them with "good morning!" or "how do you do?" Hence, any reply in the form of an answer is always "I'm fine" (followed by "and you?", as a courtesy), and never "terrible! My stomach hurts and I have a splitting headache!"

Of course, if you are talking with a friend who truly inquires about your wellbeing, and you feel like sharing with them, by all means, do so. My answer is not meant to prevent that, only to allow you the dignity of a polite answer for passers-by that does not require you to spill your guts.

Good luck!

I believe this rule only applies to U English nowadays. It was probably more widespread in the past. As far as I know, it is still de rigueur in certain circles; that is, when someone asks how do you do, you say it back; but it will now rarely be asked any more, and so the occasion for saying it back won't arise either. Perhaps this shibboleth does not extend to America; I wonder how U and non-U language work there, as I am sure they must exist in some form in nearly every culture.

I have spoken with some foreigners here who were specifically taught U Dutch and kept wondering why most people used different words. I think language learners should at least be explained the difference before being taught one variant or the other, both of which can be equally useful—that is, if their teacher is good.

Many of my non-native-English-speaking friends have also said they were taught the same as you, but with the same principle applying to all similar greetings -- "How's it going?", "How are you?", et al.

I find that bizarre. As an American English speaker, I am always taken by surprise if someone answers my "how do you do?" with another "how do you do?". While I don't always care much about their response, I do feel that I am actually asking a question to open up conversation via responses like, "Actually I just came down with a cold," or, "Great! My school application was accepted today!".

I agree with you, I expect a real response, even if it's, "Fine, you?" 90% of the time.