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We first review several recent results on rotational elastic wave modes propagating in periodic granular crystals. By considering periodically arranged elastic spheres and accounting for various contact rigidities (longitudinal, shear, bending, torsional), the existence of rotational and coupled rotational-transversal elastic waves are theoretically predicted and experimentally shown. Second, we particularly focus on the purely hysteretic rotational wave propagation in a granular chain made of magnetic spheres. The nonlinear distortion of pulses is analyzed and among several interesting effects, strong nonlinear attenuation is demonstrated.