  We present a mechanism for deterministic control of the Bloch chirality in
magnetic domain walls and skyrmions, originating from the interplay between an
interfacial Dzyaloshinskii$-$Moriya interaction (DMI) and a perpendicular
magnetic field. Although conventional interfacial DMI favors chiral N\'eel
skyrmions, it does not break the energetic symmetry of the two Bloch
chiralities in mixed Bloch$-$N\'eel skyrmions. However, the energy barrier to
switching between Bloch chiralities does depend on the sense of rotation, which
is dictated by the direction of the driving field. Our analytical and
micromagnetic analyses of steady-state Dzyaloshinskii domain wall dynamics
culminates in a switching diagram akin to the Stoner$-$Wohlfarth astroid,
revealing the existence of achiral, chiral, and precessional Bloch regimes; we
further show that in view of established energetic models, skyrmions should
exhibit the same fundamental behavior. Finally, we discuss a recent theory of
vertical Bloch line$-$mediated deterministic Bloch chirality in the
precessional regime and extend these arguments to lower driving fields. This
work establishes that the Bloch chirality of magnetic solitons can be switched
on demand with applied fields, as indicated by the Dzyaloshinskii astroid.
