There are continuum many clones on a three-element set even if they are considered up to emph{homomorphic equivalence}. The clones we use to prove this fact are clones consisting of emph{self-dual operations}, i.e., operations that preserve the relation ${(0,1),(1,2),(2,0)}$. However, there are only countably many such clones when considered up to equivalence with respect to emph{minor-preserving maps} instead of clone homomorphisms. We give a full description of the set of clones of self-dual operations, ordered by the existence of minor-preserving maps. Our result can also be phrased as a statement about structures on a three-element set, ordered by primitive positive constructability, because there is a minor-preserving map from the polymorphism clone of a finite structure $mathfrak A$ to the polymorphism clone of a finite structure $mathfrak B$ if and only if there is a primitive positive construction of $mathfrak B$ in $mathfrak A$.
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