One of the things we do not understand from the perspective of spontaneous symmetry breaking is the “handedness” of the universe. Some particles have a sense of “handedness” associated with the orientation of their spin relative to their motion. This breaks the parity symmetry (reflection off a mirror). However, this looks like the breaking of symmetry “by hand,” meaning that the particles never had it in the first place. A more elegant solution would be for the symmetry to be in place originally and then disrupted through a natural process. We hope that in a complete theory, such as string theory, handedness will be spontaneously broken, as opposed to the more contrived situation in which symmetry is artificially broken by hand.
This is somewhat reminiscent of the puzzle involving four cities at the corners of a square or rectangle, where the highways have already been built. The asymmetry in that problem is due to the constraints on the budget, which corresponds in the physical universe to the constraint of low energy.
