This paper surveys the formalism and applications of the postclassical turbulence mechanics (PCTM) grounded on the characterization of turbulent flow field in infinitesimal surroundings of the flow field points besides the flow velocity at these points also by the curvature of the velocity fluctuation streamlines passing these points. The PCTM applies this step to found the turbulence split into the orientated and the non-orientated constituents. The split specifies the competence of the classical turbulence mechanics (CTM) to the description of the non-orientated turbulence constituent and delegates the description of the orientated turbulence constituent (in the spirit of the theory of micropolar fluids) to the equation of moment-of-momentum. The concurrent presence of the orientated (relatively large scale) and the non-orientated (relatively small scale) turbulence constituents enables to compile the CTM and the conception of L. F. Richardson and A. N. Kolmogorov about the cascading turbulence (RK conception) within a conjoint formalism. The compilation solves the classical conflict between the CTM and the RK conception, though evinces a conflict of another type characterized as paradigmatic.