A spiral

Let $\mathbb{O}$ be the countable set of all infinite rational number sequences which are ultimately but not identically zero, and let $H$ contain those elements in $\mathbb{O}$ whose last nonzero element is positive. With respect to $\mathbb{O}$, define $H^\complement$. Then $f(x) = -x$ is a bijection from $H$ to $H^\complement$.

Also, $H$ looks like a spiral. To see that, require the last nonzero element to be the first, then the second, and then the third one, and watch this sequence of subsets twist into a new dimension at each step.

That's really something! A spiral whose reflection in the origin is its complement.