  Hydrodynamic helicity signatures the parity symmetry breaking, chirality, of
the flow. Statistical hydrodynamics thus respect chirality, as symmetry
breaking and restoration are key to their fundamentals, such as the spectral
transfer direction and its mechanism. Homochiral sub-system of
three-dimensional (3D) Navier-Stokes isotropic turbulence has been numerically
realized with helical representation technique to present inverse energy
cascade [Biferale et al., Phys. Rev. Lett., {\bf 108}, 164501 (2012)]. The
situation is analogous to 2D turbulence where inverse energy cascade, or more
generally energy-enstrophy dual cascade scenario, was argued with the help of a
negative temperature state of the absolute equilibrium by Kraichnan. Indeed, if
the helicity in such a system is taken to be positive without loss of
generality, a corresponding negative temperature state can be identified [Zhu
et al., J. Fluid Mech., {\bf 739}, 479 (2014)]. Here, for some specific chiral
ensembles of turbulence, we show with the corresponding absolute equilibria
that even if the helicity distribution over wavenumbers is sign definite,
different \textit{ansatzes} of the shape function, defined by the ratio between
the specific helicity and energy spectra $s(k)=H(k)/E(k)$, imply distinct
transfer directions, and we could have inverse-helicity and forward-energy dual
transfers (with, say, $s(k)\propto k^{-2}$ resulting in absolute equilibrium
modal spectral density of energy $U(k)=\frac{1}{\alpha +\beta k^{-2}}$, exactly
the enstrophy one of two-dimensional Euler by Kraichan), simultaneous forward
transfers (with $s(k)=constant$), or even no simply-directed transfer (with,
say, non-monotonic $s(k) \propto \sin^2k$), besides the inverse-energy and
forward-helicity dual transfers (with, say, $s(k)=k$ as in the homochiral
case).
