I am referring to a set of words that wouldn't make sense if one word or the other was omitted. Like barbershop quartet, or Cyber Security. What do you exactly call this set of words?
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A collocation is also an option, though the meaning is broader than requested.
This phenomenon is called a
which is both the name of the kind of bird and the name of the linguistic phenomenon because there is no such thing as a petrel of any other kind other than 'stormy'.
There are a number of examples here
It is also mentioned here at ELU.
The phenomenon could also be seen in single words which are formed from multiple morphemes, such as 'disgruntled', for which one of them never appears alone. Here is a Language log posting that is a quote of a passage intentionally filled with examples of these uncommonly exposed roots (mostly negatively prefixed words that have had the prefix removed).
From either of these lists, I don't see any examples where -neither- part stands alone.
The label for the concept that you are looking for is a
a phrase, collocation, locution (a sequence of words) that has more meaning to it than the combination of meanings of its constituents. That is, the literal meaning of its parts does not give the meaning of the entire phrase. There is a discussion here of the difference between a plain old expression and an idiom , and on set phrase.