We present a nonlinear macroscopic model in which nematic side-chain liquid single crystal elastomers are understood as materials that show two preferred directions. One of the two directions is connected to the director of the liquid crystalline phase and the other one becomes anchored in the polymer network during the procedure of synthesis. The specific properties of the materials arise from the coupling between these two preferred directions. We take into account this coupling via the variables of relative rotations between the two directions. For this purpose, we have extended the variables of relative rotations to the nonlinear regime. In addition, we generalize the concept in such a way that it can also be used for the description of other systems coupling two preferred directions. In order to test our picture, we compare its predictions to the experimental observations on nematic monodomain elastomers. As a result, we find that our model describes the nonlinear strain-induced director reorientation and the related plateaulike behavior in the stress-strain relation, which are characteristic of these materials. In addition, our model avoids the unphysical notion of a vanishing or small linear elastic shear modulus. Finally, we demonstrate that ordinary nonlinear elastic behavior of the materials, i.e., not connected to any reorientation of the director field, also plays an important role in the appearance of the stress-strain curves and must be taken into account.