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We report on the properties of nonpolar a-plane (Zn,Mg)O/ZnO quantum wells (QW) grown by molecular beam epitaxy on r plane sapphire and a plane ZnO substrates. For the QWs grown on sapphire, the anisotropy of the lattice parameters of the (Zn,Mg)O barrier gives rise to an unusual in-plane strain state in the ZnO QWs, which induces a strong blue-shift of the excitonic transitions, in addition to the confinement effects. We observe this blue-shift in photoluminescence excitation experiments. The photoluminescence excitation energies of the QWs are satisfactorily simulated when taking into account the variation of the exciton binding energy with the QW width and the residual anisotropic strain. Then we compare the photoluminescence properties of homoepitaxial QWs grown on ZnO bulk substrate and heteroepitaxial QWs grown on sapphire. We show that the reduction of structural defects and the improvement of surface morphology are correlated with a strong enhancement of the photoluminescence properties: reduction of full width at half maximum, strong increase of the luminescence intensities. The comparison convincingly demonstrates the interest of homoepitaxial nonpolar QWs for bright UV emission applications.