If somebody says "I do not smoke", you might reply "Neither do I" or "nor do I". Note that we add do when we are expressing negatives, and the word order changes next to "Neither" or "nor". I don't know why we change the word order- it sounds very old-fashioned- but we still do. Here is an NGram that shows that "Neither do I" is the only form in common use.
With neither and nor you must have exactly the same stuff in both parts: you can do it the long way and have a complete phrase, or the short way and have just one word. Here is the long way, with a complete phrase in both parts:
Neither do I belong to "classA" nor do I belong to "classB".
And here is the short way, with just a single word in both parts:
I belong to neither "classA" nor "classB".
Note that there is no do: this is because "belong" is outside the neither/nor, so it's not negative. If we moved the negative to the start of the sentence, it would be required:
I do not belong to either "classA" or "classB".