Using a high-sensitivity torsional oscillator (TO) technique, we mapped the rotational and relaxational dynamics of solid helium-4 (He-4) throughout the parameter range of the proposed supersolidity. We found evidence that the same microscopic excitations controlling the torsional oscillator motions are generated independently by thermal and mechanical stimulation. Moreover, a measure for the relaxation times of these excitations diverges smoothly without any indication for a critical temperature or critical velocity of a supersolid transition. Finally, we demonstrated that the combined temperature-velocity dependence of the TO response is indistinguishable from the combined temperature-strain dependence of the solid's shear modulus. This implies that the rotational responses of solid He-4 attributed to supersolidity are associated with generation of the same microscopic excitations as those produced by direct shear strain.