Natural convection by combined heat and mass transfer with opposing horizontal heat and solute gradients has been investigated in an anisotropic porous cavity using the Darcy model. The porous medium is assumed to be both hydrodynamically and thermally anisotropic. The principal directions of the permeability tensor are taken oblique to the gravity vector, while those of thermal and solutal diffusivity coincide with horizontal and vertical coordinate axes. Special attention is given to understand the effect of anisotropic parameters on the existence of unsteady permanent oscillations and multiple steady-state solutions. From the study of analytical solutions, which can be regarded as a verification of the numerical results, simultaneously, it is found that there exists an interval of buoyancy ratio, I-NM, depending on the parametric values, in which multiple solutions exist. For the unsteady case a similar interval, I-NO, for the buoyancy ratio has been observed numerically, in which permanent oscillations exist. Periodicity of the oscillation changes drastically by changing the permeability of the medium. The results indicate that the Maximum I-NM and I-NO interval are attained at an orientation angle of theta = 45degrees. The local direction of the flow changes because of the variation in the extent of the thermal and concentration layers, the opposite buoyant mechanism, and anisotropic parameters.