  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 $\gamma$. 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 expected to cross lesser elongated TSD bands
at the highest spins. Its transitional quadrupole moment $Q_t \approx 10.5$\,eb
is close to the measured value of $\sim$11\,eb; hence, it is a candidate for
the structure observed in experiment.
