Let G(k,n) be the k'th order jet group in n variables which consists of the set of k-jets of local diffeomorphisms of R(n) fixing the origin under the operation of composition. In coordinates, this group operation can be written explicitly using the chain rule.
Now consider G(3,1) and G(2,1) and the obvious projection homomorphism from G(3,1) onto G(2,1) induced by jets. This projection splits, that is, there is an injective homomorphism which imbeds G(2,1) into G(3,1). Identifying the image of this injection with G(2,1), we can form the left (say) coset space G(3,1)/G(2,1). Now the expression for the Schwarzian derivative defines "coordinates" on G(3,1)/G(2,1). The details of these computations can be found on pages 152-153 of the book "An Alternative Approach to Lie Groups and Geometric Structures". This construction generalizes to arbitrary dimensions and is a very special case of defining geometric structures as explained on pages 174-182 of this book.
To summarize here, these splittings arise from homogeneous spaces and define "geometric connections" which are not necessarily connections in the classical sense (keeping in mind that connections can be defined on general principal and vector bundles and are essentially topological objects). Furthermore, these splittings are built into the definition of a geometric structure and therefore there is no need to search for a "special connection" suitable for some geometric structure. Consequently, the magical powers of the Scwarzian derivative is a special case of the magical powers of "connections".
As an interesting detail, the obvious left action of G(3,1) on G(3,1)/G(2,1) gives the transformation rule of the Schwarzian derivative found in classical textbooks. In affine case (the action of G(2,n) on G(2,n)/G(1,n)), for instance, we get the well known transformation rule of the "connection components" on the tangent bundle which is historically the starting point of the theory of connections on vector bundles! The action of G(1,n) on G(1,n)/G(0,n) = G(1,n) gives absolute parallelism studied in detail in this book.
