Some categories are self-dual in ways not obvious from their definitions.
One good example is Pontryagin duality, which states that the category of locally compact Hausdorff abelian groups is self-dual, via the taking of continuous character groups.
Another is Connes' cyclic category. It is not obvious that this particular melding of the simplex category (of nonempty finite ordinals) and cyclic groups would result in a self-dual category, and in fact this property would fail if the definition were tweaked just slightly (say by working with all finite ordinals).