"Bilateral symmetry is great for this: You have an advantage over predators since it's equally likely you'll go one way vs. the other, rather than having an obvious preference for, say, left turns."
I'm pretty sure that it's very common for prey animals like fish to actually prefer turning one direction over the other to escape and that many predators have actually adapted to this behavior, preferring to strike from a position where the prey animal will flee closer to them.
"The brain is rather bilaterally symmetric itself, and quite redundant. You might have noticed the slot in the middle?"
The brain is actually divided into two parts doing different kinds of work for the sake of efficiency. Experiments have shown that doing so results in a quicker response for prey animals reacting to danger. When the brain halves are not so specialized/divided the prey animal spends far more time frozen in position when confronted by danger.
Interestingly the same design also causes the directional preference in fleeing behavior, meaning that it's better for survival to be fast rather than random.