  We report that spin supercurrents in magnetic superconductors and
superconductor/ferromagnetic insulator bilayers can induce the
Dzyaloshinskii-Moriya interaction which strength is proportional to the
superconducting order parameter amplitude. This effect leads to the existence
of inhomogeneous parity-breaking ground states combining the chiral magnetic
helix and the pair density wave orders. The formation of such states takes
place via the penetration of chiral domain walls at the threshold temperature
below the superconducting transition. We find regimes with both the single and
the re-entrant transitions into the inhomogeneous states with decreasing
temperature. The predicted hybrid chiral states can be found in the existing
structures with realistic parameters and materials combinations.
