Brain symmetry

I was thinking last night about symmetry in nature, and what biologists and physicists and mathematicians understand about it. Maybe there are natural elements that appear to exhibit symmetry (e.g., a leaf is split down the middle), but the cellular composition is actually much more complex. (Although something could be symmetrical and complex. I wonder what the opposite of symmetrical is, besides asymmetrical. Random is perhaps going too far.) Anyway, I was wondering how deep the symmetry goes. If it does go deep and it is a feature of organic structure, I wonder if it can then be a sort of lens that organisms can be seen through. So if there is an organism (or organ) that appears not to demonstrate a lot of symmetry, I wonder if it could be examined with specific symmetry parameters in mind, and then that same organism would reveal a symmetrical composition that had been previously unnoticed. Specifically, the brain. I know there's the right and left hemispheres, but I wonder how deep symmetry goes in the brain, in terms of structure but also nerve patterns. It would be interesting to explore the latest research on specific areas being sites for specific cognitive functions (e.g., left hemispehere: language) in terms of symmetry, especially because if other parts of the brain also are mildly activated during a cognitive task, they could be seen as part of the pattern. Of course, maybe even if symmetrical characteristics are common to organisms, they are also common to non-organisms, like raindrops and rocks. But that doesn't mean that the questions couldn't be tested anyway.