Here 20 cylinders intersect with axes going
through opposite corners of the following polyhedra

It is an edge-truncated cube including the hexagon face
centers except for a belt. A truncating degree for equal length edges is almost
perfect to get most isotropic oriented cylinder axes.

With radius 1 the surface of the intersection
is

12.60883195

and its volume is a third of the surface.

The intersection has 512 facets.