Here 17 cylinders intersect with axes going through opposite corners of the following polyhedra
It is not easy to find polyhedra with 17 pairs of opposing corners distributed regularily. This one is derived from the dodecahedra (20 corners) where 6 octohedra- and 8 cube-corners were added and relaxed. See scetch at the bottom.
With radius 1 the surface of the intersection is
and its volume is a third of the surface.
The intersection has 388 facets.
The construction of the 34 corner polyhedra: