  We investigate how to engineer an antisymmetric spin-split band structure
under spin density waves with finite ordering wave vectors in centrosymmetric
systems without the relativistic spin-orbit coupling. On the basis of a
perturbative analysis for the spin-charge coupled model in centrosymmetric
itinerant magnets, we show that nonzero chiral-type bilinear and biquadratic
spin cross products in momentum space under the magnetic orderings are related
to an antisymmetric spin polarization in the electronic band structure. We
apply the derived formula to the single-$Q$ cycloidal spiral and double-$Q$
noncoplanar states including the meron-antimeron and skyrmion crystals. Our
results present a clue to realize a giant antisymmetric spin splitting driven
by magnetic phase transitions in the centrosymmetric lattice structures without
the spin-orbit coupling.
