  We study the turbulent regime of chiral (magneto)hydrodynamics for charged
and neutral matter with chirality imbalance. We find that the chiral
magnetohydrodynamics for charged plasmas possesses a unique scaling symmetry,
only without fluid helicity under the local charge neutrality. We also find a
different type of unique scaling symmetry in the chiral hydrodynamics for
neutral matter with fluid helicity in the inertial range. We show that these
symmetries dictate the self-similar inverse cascade of the magnetic and kinetic
energies. Our results imply the possible inverse energy cascade in
core-collapse supernovae due to the chiral transport of neutrinos.
