I'm surprised that nobody mentioned the Tanaka-Webster connection of a strictly pseudoconvex CR spherical manifold yet.

A strictly pseudoconvex pseudo-Hermitian manifold is a triplet $(M,\theta,J)$ with $\theta$ a contact form on $M$ and $J$ an integrable almost-complex structure on $H=\ker \theta$, such that the Levi form $d\theta|_{H\times H}(\cdot,J\cdot)$ is definite positive.
On such a manifold, there exists a unique connection $\nabla^{\theta}$, called the Tanaka-Webster connection, which parallelises $H$, $J$, $X_{\theta}$ the Reeb vector field of $\theta$, and whose torsion $T^{\theta}$ satisfies
$$
\forall Y,Z \in H,\quad T^{\theta}(Y,Z) = d\theta(Y,Z)X_{\theta} \quad \text{and} \quad T^{\theta}(X_{\theta},JY) + JT^{\theta}(X_{\theta},Y)=0.
$$
The CR-sphere $S^{2n+1}$ and the Heisenberg space $H^{2n+1}$, endowed with their standard contact forms, are examples of strictly pseudoconvex pseudo-Hermitian manifolds whose curvatures $R^{\theta}$ identically vanish, but it is clear that their torsion do not.
More generally, a CR manifold which is locally CR-diffeomorphic to one of these space has the same property: they are called CR-spherical.

In this sense, the whole geometry of spherical manifolds is contained in the torsion.