Imagine a sphere with a vertical infinite line tangent to the sphere's equator. We call this infinite line a generator of the cylinder for it generates the shape. Let me stress that this line never touches the central axis. If I attached another generator to the sphere in such a way that it's a mirror image of the one before, you'll agree with me that we would get an exact hyperboloid before. It doesn't really matter which group of generators we picked to be direct or inverse.
Every direct generator meets every inverse generator, and reciprocally.
We have a hyperboloid depicted in light gray direct and inverse generators here. I've chosen a horizontal cutting plan to make things simpler. The lines at each corner of the plane are its normal vectors. The intersection of the cutting plane and the hyperboloid turns out, not surprisingly, to be a circle. These six pink points on the curve are the vertices of our conic hexagon.