Bifurcated magnetohydrodynamic (MHD) equilibrium states are computed for ITER hybrid scenario and RFX-mod SHAx configurations with very flat or reversed core magnetic shear conditions. In the ITER studies, the minimum inverse rotational transform qmin is near unity, while for RFX-mod it is 1/8. Two equilibrium states are obtained: one is axisymmetric, the other displays a 3D helical core. In tokamak devices, the structure resembles a saturated ideal MHD internal kink mode. In the reversed-field pinch, the structure is seven-fold toroidally periodic. The equilibrium magnetic field spectrum in the Boozer coordinate frame is calculated in both the ITER and RFX-mod configurations and the implications are discussed. The RFX-mod equilibria are strongly unstable to external ideal MHD kink modes, which become stabilized with a closely fitting conducting shell when the equilibrium state has a weak reversed core shear. It is marginally unstable with a monotonic q-profile. Unstable modes are driven by the Ohmic current, with pressure and Pfirsch–Schlüter currents having a very weak effect. The external kink mode spectrum is dominated by coupled m = 1, n = 6 and m = 2, n = 13 Fourier components, which revert to m = 1, n = 8 and m = 2, n = 15 terms with a conducting wall in proximity to the plasma–vacuum interface.