3d cursor orientation on 'Snap cursor to selected'

Now that the 3D cursor has an orientation, it would be great if the orientation also snapped to your selection when using ‘snap cursor to selected’. Not all selections have an obvious orientation, of course, but it could be worked out in a similar way to the direction of extrusion, which seems to be an average of the normals of any selected faces.

This feature would also be more handy if the cursor gave a visual indication of its orientation. Something like: image
or image