The structural and magnetic properties of a series of ordered double perovskites with the formula Sr(2)Co(1-x)Mg(x)TeO(6) (x = 0.1, 0.2 and 0.5) are investigated by X-ray diffraction, low temperature neutron diffraction, electron paramagnetic resonance and magnetic susceptibility. The progressive substitution of the paramagnetic Co(2+) high spin ion by the diamagnetic Mg(2+), of about the same size, induces changes in the room temperature crystal structure, from a distorted P2(1)/n phase for the undoped Sr(2)CoTeO(6) oxide to the I4/m of the end member (Sr(2)MgTeO(6)). These perovskites experience structural transitions on heating, the temperature at which the transitions occur being smaller as x increases. The novel approach of mode-crystallography is used for the analysis. All oxides show antiferromagnetic exchange interactions between Co(2+) ions but the long range antiferromagnetic order is not achieved for the phase with x = 0.5. The low temperature neutron diffraction data have been evaluated using a full symmetry analysis. Results are consistent with an unquenched orbital contribution of a high spin Co(2+) ion.