Answering a question of Conway and Guy (SIAM Rev. 11:78-82, 1969), Langi (Bull. Lond.
Math. Soc. 54: 501-516, 2022) proved the existence of a monostable polyhedron with
n-fold rotational symmetry for any n = 3, and arbitrarily close to a Euclidean ball.
In this paper we strengthen this result by characterizing the possible symmetry groups
of all mono-monostatic smooth convex bodies and convex polyhedra. Our result also
answers a stronger version of the question of Conway and Guy, asked in the above paper
of Langi.
