In this work, Marwell-Hertz electrodynamics (MHE), valid for nonuniform motions as they occur in the physical reality and hold for the noninertial reference frame of our laboratory but at small velocities only, is extended at relativistic velocities. The new theory, called Hertz's relativistic electrodynamics (HRE), is completely independent and built-up in a completely different way than Einstein's special relativity (ESR). HRE, a coordinate-free formulation, does not need postulates, but confirms the constancy principle of the speed of light in a vacuum. All experiments of first and second order in v(2)/c(2) are correctly interpreted. To this theory a Hertzian kinematics and dynamics are associated. HRE with its corresponding mechanics form Hertz's special relativity (HSR) as a theory complementary to ESR. According to the principle of complementarity and neglecting the gravitational effects, the extended special relativity (ExSR) is a double-faced theory which becomes either ESR when the motion is inertial or HSR when the motion is noninertial. The complementarity of both theories assumes that the two descriptions cannot be employed for the same motion, being mutually exclusive. Consequently, to every statement of one of the ExSR, a complementary statement of the other ExSR corresponds. The completeness of ESR with HSR ensures an extended view over the relativity in our physical world.