I'm trying to think of a tag for "either, neither, nor", in the example below, for nor:
- Our neighbours aren’t very polite, and _____ particularly quiet! (answer: nor are they)
How a grammar book would categorize it?
In my dialect, "and nor" doesn't make any sense. Here are some examples that do make sense to me:
Here I regard the "and" as a coordinating conjunction and the "neither" as an adverb.
Here there is no "and". The coordinating conjunction is "nor". Other common coordinating conjunctions are "but", "or" and "yet".
We can still regard "neither" as and adverb and regard "nor" as a coordinating conjunction. However, the neither/nor pair is more commonly described as a "correlative conjunction". Other common correlative pairs include either/or, both/and, not/but and not-only/but-also.
The particular example you give:
*"Our neighbours aren’t very polite, and nor are they particularly quiet!"
The words "and nor" are coordinating conjunctions. The word "and" is redundant here. This kind of redundancy is reasonable common in speech, though would not be typical of written English)
The words "or" and "nor" is nearly always a conjunction, except in technical use (the NOR logical gate). On the other hand, "either" and neither can function like a determiner ("neither answer was correct"), a pronoun ("There were two answers, but neither was correct") or a special type of conjunction (used with nor) "Neither 'dog' nor 'cat' is correct".
The last of these deserves particular attention, since it is an example of a single grammatical function being formed by two words. The word "or" can be both an "inclusive or" or an "exclusive or". (see this ELU question) The word "either" pairs with "or" to form an unambiguous exclusive or "I will either buy a hat or go to the cinema." is unambiguously exclusive. The word "either" pairs with the word "or" to form a single function: "conjunction". Other languages use a single word or "either...or" (Basque has ala, Finnish has vai, and there are other examples) So the short answer to your question is "no" the function of "either" doesn't have a name, since it is not complete without "or".
Until you use a word in a sentence it doesn't have a part of speech.