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Hydrodynamic helicity signatures the parity symmetry breaking, chirality, ofthe flow. Statistical hydrodynamics thus respect chirality, as symmetrybreaking and restoration are key to their fundamentals, such as the spectraltransfer direction and its mechanism. Homochiral sub-system ofthree-dimensional (3D) Navier-Stokes isotropic turbulence has been numericallyrealized with helical representation technique to present inverse energycascade [Biferale et al., Phys. Rev. Lett., {\bf 108}, 164501 (2012)]. Thesituation is analogous to 2D turbulence where inverse energy cascade, or moregenerally energy-enstrophy dual cascade scenario, was argued with the help of anegative temperature state of the absolute equilibrium by Kraichnan. Indeed, ifthe helicity in such a system is taken to be positive without loss ofgenerality, a corresponding negative temperature state can be identified [Zhuet al., J. Fluid Mech., {\bf 739}, 479 (2014)]. Here, for some specific chiralensembles of turbulence, we show with the corresponding absolute equilibriathat even if the helicity distribution over wavenumbers is sign definite,different \textit{ansatzes} of the shape function, defined by the ratio betweenthe specific helicity and energy spectra $s(k)=H(k)/E(k)$, imply distincttransfer directions, and we could have inverse-helicity and forward-energy dualtransfers (with, say, $s(k)\propto k^{-2}$ resulting in absolute equilibriummodal spectral density of energy $U(k)=\frac{1}{\alpha +\beta k^{-2}}$, exactlythe enstrophy one of two-dimensional Euler by Kraichan), simultaneous forwardtransfers (with $s(k)=constant$), or even no simply-directed transfer (with,say, non-monotonic $s(k) \propto \sin^2k$), besides the inverse-energy andforward-helicity dual transfers (with, say, $s(k)=k$ as in the homochiralcase).

Note on specific chiral ensembles of statistical hydrodynamics: "order function" for transition of turbulence transfer scenarios