Invariance considerations are employed to write down constitutive equations governing the propagation of electromagnetic waves in isotropic materials with a centre of symmetry which are subject to a static deformation. It is assumed that the dielectric displacement and magnetic induction vectors are linear functions of the electric and magnetic field intensities, respectively, but are general polynomial functions in the quantities which specify the deformation. The theory is employed to examine propagation along circular cylindrical rods in torsion. Rotating waves are produced whose speed of propagation and rate of rotation depend upon the magnitude of the deformation and the properties of the material. The nature of these waves is examined for the general case where there is no restriction either upon the amount of torsion or upon the magnitude of the effect. When the amount of torsion, or the dependence of the effect upon deformation is small, solutions can be obtained based upon those for the propagation of waves in undeformed materials.