  Two related questions are discussed. The first is when reflection symmetry in
a finite set of $i$-dimensional subspaces, $i\in \{1,\dots,n-1\}$, implies full
rotational symmetry, i.e., the closure of the group generated by the
reflections equals $O(n)$. For $i=n-1$, this has essentially been solved by
Burchard, Chambers, and Dranovski, but new results are obtained for $i\in
\{1,\dots,n-2\}$. The second question, to which an essentially complete answer
is given, is when (full) rotational symmetry with respect to a finite set of
$i$-dimensional subspaces, $i\in \{1,\dots,n-2\}$, implies full rotational
symmetry, i.e., the closure of the group generated by all the rotations about
each of the subspaces equals $SO(n)$. The latter result also shows that a
closed set in $\mathbb{R}^n$ that is invariant under rotations about more than
one axis must be a union of spheres with their centers at the origin.
