  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.
