Symmetry breaking: The current "paradigm" that has been proved to be so useful in high-energy physics (i.e. what we may pragmatically call "fundamental physics") is that at the small scale some exact symmetry is there, and then (at a large scale) the symmetry may be broken. Effective theories (i.e. theories that are less "fundamental") must be formulated to describe physics after a symmetry breaking: this kills the idea of "reductionism" since (because of symmetry breaking) theories at different scales have little in common and it's very difficult to derive them from the "fundamental" one (i.e., the most symmetric one available on the market at a certain moment).
This is well explained in a wonderful paper: I suggest you to read "More is Different" by <PERSON>, where there is a beautiful and clear discussion of how emergent physics is related to symmetry breaking. 675
Also, this question is extremely interesting as it remarks that the "symmetry of objects" should be distinguished from the "symmetry of laws".
The opposite of symmetry breaking: What you are asking is the opposite of what <PERSON> describes in his paper (a sort of "symmetry restoration", or "emergent symmetry"). Apart from the idea of "emergent symmetry" (see this and this questions), another possible interpretation of what you are asking for is that of universality class, where emergent phenomena in large systems behave in the same way despite some microscopic details may be different. Note that the idea of universality class is not really that of "symmetry restoration", but just something philosophically related in the sense that "details are not important". After all, symmetry tells us that a particular operation can be performed without changing some result: the idea of universality class is that you can add some kind of "irrelevant" interactions and this does not affect the large-scale behaviour of the system at criticality. In this sense, it may be considered a sort of emergent symmetry: all the systems belonging to the same class are the same.
Alternatively, there are accidental symmetries, see this or Wiki, but they are more the exception than the rule (they need fine-tuning and are not exact), so I think that this concept does not really fit your question. 682