Back when I was studying electromagnetism and Maxwell's equations, our teacher told us a quote. I can't recall it exactly, but the meaning was roughly the following:
Symmetry in a problem is useless if you don't have the means to exploit it.
(by the way, I would be delighted if a nice soul could provide the source for it.)
It makes sense in the context of electromagnetism: the effect of symmetries in the initial condition is not as simple as one might naively think. For instance, a planar symmetry for the charges yields a planar symmetry for the electric field, while a planar symmetry for the current yields an antisymmetry for the magnetic field. Hence, the effect of a given symmetry in the initial conditions depends on the properties of the equations.
I was later quite surprised when I learned of some much more striking examples. The first one which comes to mind is the following:
What is the shortest graph which connects the vertices of a square?
The first reflex of most people would be to look at graphs which have the same symmetry as the square ($D_4$). That's an error. The solutions exhibit some degree of symmetry ($D_2$), but less than the square!
However, I don't know any other nice examples for which the solution is less symmetric than the problem (except perhaps sphere packings, but that's less surprising). I think it would be nice to have a list as diverse as possible, both to hone my intuition and to provide counter-examples to my students. And, frankly, because this kind of phenomenon is quite fun.