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' <a href="https://en.wikipedia.org/wiki/Cyclic_category">cyclic category</a>. 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).