I was in a conversation with thoughtful and confrontational folks -- technical people --- sometime ago, and they asked me in a provoking way, "what is the big deal with Symmetry, why do we care?" This was obviously meant to skewer me trying to defend the math, science and beauty of symmetry. Humans make symmetric objects (airplanes, say), nature makes symmetric objects (leaves of trees, say), and symmetry abounds in the particles of the Universe. But the conversation was in a context where I had to reach for something more down to Earth. So, I asked the men in the group, "Say you need to go to the restroom and you hurry, but find the women's. Where will you look for the men's?" They understood the importance of symmetry. :)
ps: When I asked some of the women, "Say you need to go to the restroom and you hurry, but find the men's.
Where will you look for the women's?" Some of them said, "I will ask someone." This was smart, and it killed my argument.