Can adhd and ocd coexist? Is it in the same way mania and depression can coexist (in bipolar disorder)?
Yes, they can coexist. No, they aren't opposites. How could they be? Both ADHD and OCD affect a variety of different brain regions, and its effects are not consistent in every person. Maybe if OCD and all its symptoms were always caused by an excess of dopamine in one specific pathway in the reward system, whereas ADHD and all its symptoms was always caused by an decrease of dopamine in that same system, then they wouldn't be compatible. But this isn't the case, and it could never be the case, because the brain is more complex than that.

If mental disorders were diagnosed strictly on oppositional behaviors, then we could easily make the assertion that the opposite of being inattentive would be being hyperfocused. Therefore, nobody who has an attention deficit disorder could achieve hyperfocus, right? But we know to attribute both of these behaviors to ADHD. How can this be?

It is important to remember that mental disorders are not behaviors. OCD is marked by obsessions and compulsions, and any behaviors that may arise do so in the context of mental symptoms. It certainly follows that someone with OCD who has an obsessional focus on staying clean and organized will stay clean and organized -- in general, it is hard to not act on things that you are constantly thinking about and uncomfortable by. This is different from the child with ADHD, who may or may not have trouble staying clean and organized, because to someone who doesn't have some sort of incentive (or 'motivation'), staying clean and organized is really hard to force yourself to do on your own terms. And with ADHD, it can be all the more difficult to achieve that motivation, given the lack of dopamine and/or norepinephrine activity in certain parts of the brain.

That being said, ADHD is not a 'messiness' disorder. The behavior of an ADHD sufferer arises from a complex set of symptoms, usually related to executive dysfunction. As the name may reveal, executive functioning is an abstract yet measurable concept that encompasses a variety of higher-level processes, including regulation of focus, self-management (including keeping oneself organized), and one's capacity for paying attention to detail. Any messiness that arises from this is just an effect from the cause-and-effect relationship that is ADHD.

Likewise, OCD is not a 'neatness' disorder. OCD is marked by obsessions and compulsions. A child with ADHD may have OCD and never feel any compulsion to clean a messy room. There are many different ways that obsessions and compulsions can be demonstrated. A person with OCD may perform rituals, engage in lengthy routines, avoid certain foods, or pick at their skin. They may fear or avoid strange things for no reason at all. Many people with OCD have phobias, and these phobias may interfere with normal life, such as a fear of blood leading someone to never step foot in a doctor's office. Perfectionism and OCD are not the same thing, though perfectionism can be a quality in both.