According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) Legendre transformation (disguised comp. inversion) for cyclic quadratic operads.
Chapoton, Vallette, Loday, and others have used binary trees to characterize these relationships. There are numerous combinatoric structures related to these operations, including permutohedra and mappings of weighted surjections for forming the reciprocal of exponential generating series, and Stasheff polytopes (type A associahedra) for compositional inversion of ordinary generating series.
What combinatoric/geometric structures, do you feel, give you the most enlightening insights on the relationships between these inversions and Koszul duality? (with some comment on how/why)