The Schwarzian derivative encodes the adjoint action of the Bott-Virasoro group. One version of it is ${Diff}\_{\mathcal S}(\mathbb R)\times \mathbb R$(here $\mathcal S$ stands for "rapidly falling towards the identity") with multiplication$$\binom{\phi}{\alpha}.\binom{\psi}{\beta} = \binom{\phi\circ\psi}{\alpha+\beta+c(\phi,\psi)}$$where the Bott cocycle is: $$c(\phi,\psi) = \frac12\int\_{\mathbb R} \log(\phi'\circ \psi)\,d\log(\psi').$$This is alluded to on page 22ff of the book of Ovsienko and Tabashnikov mentioned in Viktor Protsaks answer. A short exposition along the lines of this answer is on page 55 of (here). I think that this encodes much of the magic of the Schwartzian derivative. Dualize it to the coadjoint action and note that many coadjoint orbits are of the form $Diff\_{\mathcal S}(\mathbb R)/PSL(2,\mathbb R)$ to see the projective properties. Etc.
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