This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this added structure, this leads to symmetries. One then needs to look at the original solution of the tractable case and modulo cases congruent under symmetry.
The simplest example is when we study free modules. We add a basis to make the problem tractable, reducing to linear algebra. However, when attempting to make basis-free statements of linear spaces and maps, one then must talk about matrices up to similarity, and this is the congruence relation. The change of base are symmetries.
Another example is the use of spectral sequences. The original grading of the graded ring may not be amenable to computation. So we add the structure of a filtration to introduce another grading. Again, one needds to forget this structure if one wants the original grades of the ring.
Do people have other examples of such a situation?