Exact solutions are found for an -fold rotationally symmetric, steadily rotating hollow vortex where a continuous real parameter governs its deformation from a circular shape and is an integer. The vortex shape is found as part of the solution. Following the designation 'V-states' assigned to steadily rotating vortex patches (Deem & Zabusky, Phys. Rev. Lett., vol. 40, 1978, pp. 859-862) we call the analogous rotating hollow vortices 'H-states'. Unlike V-states where all but the solution - the Kirchhoff ellipse - must be found numerically, it is shown that all H-state solutions can be written down in closed form. Surface tension is not present on the boundaries of the rotating H-states but the latter are shown to be intimately related to solutions for a non-rotating hollow vortex with surface tension on its boundary (Crowdy, Phys. Fluids, vol. 11, 1999a, pp. 2836-2845). It is also shown how the results here relate to recent work on constant-vorticity water waves (Hur & Wheeler, J. Fluid Mech., vol. 896, 2020, R1) where a connection to classical capillary waves (Crapper, J. Fluid Mech., vol. 2, 1957, pp. 532-540) is made. ©