<b>Abstract</b>— <p>The key role of the family of hydromagnetic helicity invariants in connection with the generation and maintenance of magnetic fields in geophysical and astrophysical contexts is discussed. The influence of compressibility and rotation on the turbulent mass transport in helical hydromagnetic flows is investigated using a phenomenological approach at very high Reynolds numbers. The fluctuating effects entering into the averaged MHD equations through their correlation contributions and representing the hydromagnetic turbulent stress, turbulent electromotive force and a number of other correlation functions are modeled using linear closure relations (in the absence of reflective symmetry of small-scale motions) and differential equations for four helical chiral turbulence descriptors, which are: total turbulent plasma energy, turbulent transverse helicity, turbulent residual energy and turbulent residual helicity. It is believed that the model equations for these descriptors, combined with the compressible MHD mean field equations, allow the most complete construction of a self-consistent model of the turbulent dynamo. The ultimate goal of the undertaken research is the development of models of helical hydromagnetic turbulence capable of operating effectively in the hypersonic regime.</p>