Stimulated by recent experimental discoveries, triaxial strongly deformed (TSD) states in (158)Er at ultrahigh spins have been studied by means of the Skyrme-Hartree-Fock model and the tilted-axis-cranking method. Restricting the rotational axis to one of the principal axes--as done in previous cranking calculations--two well-defined TSD minima in the total Routhian surface are found for a given configuration: one with positive and another with negative triaxial deformation γ. By allowing the rotational axis to change direction, the higher-energy minimum is shown to be a saddle point. This resolves the long-standing question of the physical interpretation of the two triaxial minima at a very similar quadrupole shape obtained in the principal-axis-cranking approach. Several TSD configurations have been predicted, including a highly deformed band, which is a candidate for the structure observed in experiment.