One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- (see below) &c&c, to be convinced that modified concepts are replete across maths, proliferating, and their diversity is likely accelerating.
Shafarevich: "it is the destiny of mathematics to expand in all directions."
This trend, coupled with the lack of standardized terminology, can make it difficult to compare results or in same cases even definitions.
It seems clear that in general a modifier term doesn't categorically reveal whether the modified concept is a specialization or generalization of the underlying concept (eg, subset versus superset, or subcategory versus supercategory). In some cases the modified concept might not bear a sub/super relation to the underlying, for exmaple, co- and op- in category theory and universal algebra (what's the relationship of universal co-algebra to algebra or co-induction to induction?).
So it appears we must be content with enumerating cases to discern the relation and then compare to see if a big picture emerges. Basic examples:
Semigroups are generalizations of groups but inverse semigroups are specializations of semigroups. (Quasicrystals are crystals - this got the Nobel - but their symmetries don't satisfy the crystal restriction theorem, eg, translation invariance, so are not groups, but might be modeled by inverse semigroups [ML]).
Quasimetrics are generalizations of metrics, but ultrametrics are specializations of the latter[VS] .
Noncommutative geometry, Connes stresses, includes commutative geometry so it is a generalization.
In the absence of an online OEIS-like database, would it be possible to crowd-source many more examples of mathematical concepts or categories noting sub/super (or other) relation to the underlying?