Ancient genome duplications laid the foundations of complex brains

No, it does not have any connection with symmetry whatsoever.

Genome duplication has happened in many animal groups. Wherever it happens, it enables the descendants to evolve into having bodies that are bigger and more complex, because each ancestral gene is replaced by multiple copies and then each copy evolves differently, becoming able to accomplish additional functions than in the ancestors.

Without a genome duplication, accumulating a similar number of genetic innovations would require a much longer time.

Another example besides the vertebrates is that the ancestor of spiders and scorpions also passed through a genome duplication event.

In animals, the genome duplications are very rare events, because the development of animal embryos is very complicated and after a genome duplication the embryos will usually fail to develop correctly and they would die.

On the other hand, in plants genome duplications and also hybridizations when a genome is doubled by combining together 2 genomes of some closely related plants, are very frequent.

Some of the most important crops have genomes that have been multiplied, either from the same species or by hybridization, i.e. which are called tetraploid or hexaploid, to mark the fact that they are doubled or tripled in comparison with the original diploid genomes. In cultivated plants, this genome multiplication has resulted in a higher productivity in comparison with their wild ancestors.

I don't think that has much to do. Symmetry is very handy in several fronts, you can code an organism geometry more easily, that same organism has an easier time learning how to navigate environments and another organism of the same species has an easier time detecting defects on it while weighing potential mating.