The continuous magnetohydrodynamic spectrum is investigated for axially symmetric equilibria in which the plasma rotates toroidally in a purely poloidal magnetic field. The flow continuum is given by a simple algebraic expression, while the Alfve´n‐ and cusp continua (which are coupled through the Coriolis force) are given by the eigenvalues of a fourth‐order non‐self‐adjoint problem in ordinary differential equations along magnetic field lines. It is shown that these continua are stable if the entropy does not increase along the field lines away from the rotation axis, but that the cusp continuum may turn unstable otherwise.