On convex bodies with rotationally symmetric planar projections

Let $n\ge 3$ and let $K\subset {\mathbb{R}}^n$ be a convex body. For a two-dimensional linear subspace $P\subset {\mathbb{R}}^n$, let $K\mathrm{\mid }P$ be the orthogonal projection of $K$ onto $P$. We prove that if for every two-dimensional subspace $P$, the planar convex body $K\mathrm{\mid }P$ has $q$-fold rotational symmetry up to translation, then for $q\ge 4$, this forces $K$ to be an Euclidean ball. The case $q=3$ is exceptional: non-spherical examples exist.