  We present a symmetry-based approach for prolate-oblate and
spherical-prolate-oblate shape coexistence, in the framework of the interacting
boson model of nuclei. The proposed Hamiltonian conserves the SU(3) and
$\overline{\rm SU(3)}$ symmetry for the prolate and oblate ground bands and the
U(5) symmetry for selected spherical states. Analytic expressions for
quadrupole moments and $E2$ rates involving these states are derived and
isomeric states are identified by means of selection rules.
