In 2008, Guichard-Wienhard showed that PSL(4,R)-Hitchin representations of surface groups are exactly the holonomies of "properly convex foliated" projective structures on unit tangent bundles of surfaces. In this talk we will discuss phenomena that emerge in this story, with an eye towards connections to other directions in projective geometry and higher Teichmüller theory. After presenting the setting, we will explain two such connections. First, the leaves of a codimension-one foliation show a new point-set topological property of the (non-Hausdorff) space of projective equivalence classes of convex domains in RP(n), which in particular answers an old question of Benzécri. Second, we describe how a codimension-two foliation is connected to a concrete and general construction that reproduces some flows that are studied on the dynamical side of higher Teichmüller theory. Part of this is based on joint work with Max Riestenberg.