This paper is an attempt to introduce methods and concepts of the Riemann–Cartan geometry largely used in such physical theories as general relativity, gauge theories, solid dynamics to fluid dynamics in general and to studying and modeling turbulence in particular. Thus, in order to account for the rotational degrees of freedom of the irregular dynamics of small-scale vortexes, we further generalize our unified first-order hyperbolic formulation of continuum fluid and solid mechanics which treats the flowing medium as a Riemann–Cartan manifold with zero curvature but non-vanishing torsion. We associate the rotational degrees of freedom of the main field of our theory, the distortion field, to the dynamics of microscopic (unresolved) vortexes. The distortion field characterizes the deformation and rotation of the material elements and can be viewed as anholonomic basis triad with non-vanishing torsion. The torsion tensor is then used to characterize distortion’s spin and is treated as an independent field with its own time evolution equation. This new governing equation has essentially the structure of the nonlinear electrodynamics in a moving medium and can be viewed as a Yang–Mills-type gauge theory. The system is closed by providing an example of the total energy potential. The extended system describes not only irreversible dynamics (which raises the entropy) due to the viscosity or plasticity effect, but it also has dispersive features which are due to the reversible energy exchange (which conserves the entropy) between micro- and macroscales. Both the irreversible and dispersive processes are represented by relaxation-type algebraic source terms so that the overall system remains first-order hyperbolic. The turbulent state is then treated as an excitation of the equilibrium (laminar) state due to the nonlinear interplay between dissipation and dispersion.