"Asymmetric" results in maths analogous to "Parity violation" of the weak force? Disclaimer: I'm not a physicist and I don't claim to be one so if I have any mistakes I’ll be glad to be corrected.
One feature of the standard model of particle physics is that the weak force is not symmetric with respect to a parity transformation (to be more specific only left handed fermions interact with the weak force).  
This is an example of a special kind of counter-intuitive result that defies a symmetry that looks extremely reasonable given our intuition about the world. This implies that the world we see when we glance at the mirror is 
"unphysical" (or at least obeys a different law for weak interaction).

I'm looking for results in maths where an obvious symmetry was
  suggested by a certain problem yet careful consideration revealed
  that the symmetry is in fact broken for certain cases.

Although there are many threads in SE about counter-intuitive results in maths I don't think my question is a duplicate. If it is indeed a duplicate or not at all appropriate I'm sorry.

For finite dimensional real vector spaces $V$ we have an "obvious" and canonical isomorphism with its bidual $V^{**}$. This "symmetry" breaks for infinitedimensional spaces.

Also in the finite case, ordinals and cardinals are "the same", but this breaks for infinites
