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We demonstrate that low-lying triplon excitations in a bilayer Heisenberg antiferromagnet provide a promising avenue to realize magnetic analogs of twisted superfluid and supersolid phases that were recently reported for two-component ultracold atomic condensate in an optical lattice. Using a cluster Gutzwiller mean-field theory, we establish that Dzyaloshinskii-Moriya interactions (DMI), that are common in many quantum magnets, stabilize these phases in a magnetic system, in contrast to the pair hopping process that is necessary for ultracold atoms. The critical value of DMI for transition to the twisted superfluid and twisted supersolid phases depends on the strength of the (frustrated) interlayer interactions that can be tuned by applying external pressure on and / or shearing force between the layers. Furthermore, we show that the strength of DMI can be controllably varied by coupling to tailored circularly polarized light. Our results provide crucial guidance for the experimental search of twisted superfluid and supersolid phases of triplons in real quantum magnets.