Is there a general rule of symmetry in mathematics and science (see details)?

I apologize for the vagueness of the question. Let me give an example or two to show what I mean.

The angles x, y and z of a triangle satisfy x + y + z = 180. Addition is symmetric in the sense that the order of the terms makes no difference, so x, y and z have indistinguishable roles in the equation. This agrees with our intuition (as well as geometric axioms and definitions) that for a global property of the angles, there should be no way of distinguishing one angle from another.

Suppose instead that the condition was 2x + y + z = 180 or x + 2y + 2z = 180 or x + y +2z = 180. The combination of conditions does not distinguish between x, y and z, but it says that one of them plays a different role in the equation, although we don’t know which one.

If you are thinking that the second case is a bit strange, consider this. From x + y + z = 180, we can say that unless x = y = z = 60, then x < 60 or y < 60 or z < 60, and similarly x > 60 or y > 60 or z > 60. This is a case like the second one, where the variables may play distinguishable roles, but there is no way of knowing from the information how the roles will be assigned.

As far as useful general equations applying to global properties are concerned, I don’t know any of the second form. Can you think of any? Could there be an alternative universe where the second type was the general rule?