"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.

In example in probablity theory would be the monty hall problem (http://en.wikipedia.org/wiki/Monty_Hall_problem). Most people seem, at least initially, to assume that there's a symmetry between the two unopened doors, and hence believe that the probability of winning is the same regardless of whether you switch doors or not. Yet it turns out that the probability of winning is higher if you switch.
Another problem, although one which deals with an extrapolation rather than a symmetry in a strict sense, is solving polynomial equations. After being presented the formula for finding roots of univariate polynomials of degrees $1$, $2$, $3$ and $4$, you'd naturally assume that there's a formula for every degree. Yet it turns out that there's no universal formula for degree $5$ and higher. 
