You might be interested in the concept of "duality" often used in e.g. theoretical physics: https://en.wikipedia.org/wiki/Duality_(mathematics)

If you want to be even more general, check out a field of mathematics called category theory (https://en.wikipedia.org/wiki/Category_theory). Its' central idea is to describe mathematical structures in terms of "objects" and their relationship with each other (called morphisms). I'm not extremely knowledgeful in this subject, but I believe mostly everything you get out of categorical theoretical treatment is too generic to be useful, but has resulted in some real results in e.g. algebraic geometry.