An addition to a mathoverflow post

We add to - generalize - this post at mathoverflow and add a couple of questions. The mathoverflow post is on central symmetry. We here talk about any symmetry:

Note: in what follows, instead of "mirror symmetry", one can think of any other symmetry.

1. Which convex solids have the property: all planar sections have mirror symmetry?

2. If a convex solid has the property: all its maximal area planar sections - sections of max area with normal in any specified direction - have mirror symmetry, what can be said about the solid?

Note: in both above questions, we can replace max area planar sections with max perimeter planar sections. One can also consider shadows instead of planar sections; and solids for which both shadows and planar sections have some specified symmetry.