The operadic butterfly is a diagram in the category of operads in vector spaces. It extends the short exact sequence relating commutative, associative and Lie operads.

$$\begin{array}{ccccc} & Dend & & & & Dias & \newline \nearrow & &\searrow & &\nearrow & &\searrow\newline Zinb & & &Ass & & & \quad Leib \newline \searrow & &\nearrow & &\searrow & &\nearrow\newline & Comm & & & & Lie & \newline \end{array}$$

Here is a paper by Loday in which some discussion can be found. As the reference explains, the Koszul duality functor, $\mathcal{O}\mapsto \mathcal{O}^!$, gives the above diagram symmetry about the vertical axis.

Is the butterfly also symmetric about the horizontal axis?

In other words, does there exist a functor $F : Operad(C) \to Operad(C)$, where $C$ is vector spaces or chain complexes, such that $F\circ F \simeq 1$, $F$ fixes the operads: $Zinb$, $Ass$ and $Leib$, and exchanges the pairs: $(Dend,Dias) \leftrightarrow (Comm,Lie)$.

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