  We consider a modified gravity plus single-scalar-field model, where the
scalar Lagrangian couples symmetrically both to the standard Riemannian
volume-form (spacetime integration measure density) given by the square-root of
the determinant of the Riemannian metric, as well as to another non-Riemannian
volume-form in terms of an auxiliary maximal-rank antisymmetric tensor gauge
field. As shown in a previous paper, the pertinent scalar field dynamics
provides an exact unified description of both dark energy via dynamical
generation of a cosmological constant, and dark matter as a "dust" fluid with
geodesic flow as a result of a hidden Noether symmetry. Here we extend the
discussion by considering a non-trivial modification of the purely
gravitational action in the form of f(R) = R - \alpha R^2 generalized gravity.
Upon deriving the corresponding "Einstein-frame" effective action of the latter
modified gravity-scalar-field theory we find explicit duality (in the sense of
weak versus strong coupling) between the original model of unified dynamical
dark energy and dust fluid dark matter, on one hand, and a specific quadratic
purely kinetic "k-essence" gravity-matter model with special dependence of its
coupling constants on only two independent parameters, on the other hand. The
canonical Hamiltonian treatment and Wheeler-DeWitt quantization of the dual
purely kinetic "k-essence" gravity-matter model is also briefly discussed.
