We consider a cosmological model in which the two major fluid components of the Universe, dark energy and dark matter, flow with distinct four-velocities. This cosmological con figuration is equivalent to a single anisotropic fluid, expanding with a four-velocity that is an appropriate combination of the two fluid four-velocities. The energy density of the single cosmological fluid is larger than the sum of the energy densities of the two perfect fluids, i.e., dark energy and dark matter, respectively, and contains a correction term due to the anisotropy generated by the differences in the four-velocities. Furthermore, the gravitational field equations of the two-fluid anisotropic cosmological model are obtained for a Bianchi type I geometry. By assuming that the non-comoving motion of the dark energy and dark matter induces small perturbations in the homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker type cosmological background, and that the anisotropy parameter is small, the equations of the cosmological perturbations due to the non-comoving nature of the two major components are obtained. The time evolution of the metric perturbations is explicitly obtained for the cases of the exponential and power law background cosmological expansion. The imprints of a non-comoving dark energy - dark matter on the Cosmic Microwave Background and on the luminosity distance are briefly discussed, and the temperature anisotropies and the quadrupole are explicitly obtained in terms of the metric perturbations of the flat background metric. Therefore, if there is a slight difference between the four-velocities of the dark energy and dark matter, the Universe would acquire some anisotropic characteristics, and its geometry will deviate from the standard FLRW one. In fact, the recent Planck results show that the presence of an intrinsic large scale anisotropy in the Universe cannot be excluded a priori, so that the model presented in this work can be considered as a plausible and viable working hypothesis.
