Without being an expert on the topic, I think it is safe to say that some symmetry in the nervous system is present throughout the entire animal kingdom -- just as bodies tend to be symmetric. For insects in particular, the brain is definitively split into two clearly distinguishable halves.

Octopuses in particular are a completely different story, they don't seem to possess a "central processing unit", but rather distribute necessary computations throughout their entire nervous system (it is kind of fascinating). Yet, since they have two eyes, you'll still find two symmetrical nuclei for processing visual information somewhere in the vicinity of the sensor system.

I think every animal from arthropods (~bugs) upward in terms of complexity have connected nervous systems with bilateral symmetry. But many simple invertebrates don't, like jellyfish.

But the mere presence of bilateral interconnectedness doesn't mean that their 'brain' conveys much information or coordinates much activity in those linnaean classess. In all invertebrates, most(?) nervous control arises locally, via clumps of neurons (ganglia). The lower the class, the larger the fraction of control arising locally from ganglia. For example, earthworms have only ganglia (IIRC).

Spiders have donut shaped brains. Source: owned a tarantula. Some animals jon't have bilaterally symmetric brains on account of having brains, like jelly fish.

Excellent question... does anyone know?