We present a mechanism for deterministic control of the Bloch chirality in magnetic skyrmions originating from the interplay between an interfacial Dzyaloshinskii$-$Moriya interaction (DMI) and a perpendicular magnetic field. Although conventional interfacial DMI favors chiral Neel skyrmions, it does not break the energetic symmetry of the two Bloch chiralities in mixed Bloch$-$Neel 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 analysis of steady-state Dzyaloshinskii domain wall dynamics culminates in a switching diagram akin to the Stoner$-$Wohlfarth astroid, revealing the existence of both monochiral and multichiral Bloch regimes. Furthermore, we discuss recent theory of vertical Bloch line$-$mediated Bloch chirality selection in the precessional regime and extend these arguments to lower driving fields. This work establishes that applied magnetic fields can be used to dynamically switch between the chiral Bloch states of domain walls and skyrmions as indicated by this new Dzyaloshinskii astroid.
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