We report the existence of persistently rotating azimuthons in media with self-focusing Kerr and absorption nonlinearities. The nonlinear loss is balanced by power influx from the peripheral reservoir stored in a slowly decaying tail of the field. The azimuthon modes are excited by a superposition of two Bessel beams with opposite vorticities, ±s, and slightly different conicities. The excited mode exhibits vorticity in its center opposite to that of the input Bessel-beam superposition, due to spontaneous inversion of the topological charge in the course of the azimuthon formation. Unlike azimuthons in loss-free media, number N of rotating intensity maxima and s are not mutually independent, being related by N=2s. The robustness of the rotating azimuthons is enhanced in comparison to similar static dissipative patterns. They can be excited in almost any transparent material, in the range of intensities for which the nonlinear absorption, induced by multiphoton absorption, is relevant. Close to the ionization threshold, the rotating azimuthons are similar to recently observed helical filaments of light in air and CS2.