We reexamined the rotation-induced effect observed in solid He-4 by using a rigid two-frequency torsional oscillator (TO). The previous rotation experiments reported the rotation-induced suppression of the "nonclassical" TO response that was interpreted as evidence of irrotational bulk superfluidity in solid He-4. However, the experiment employed a nonrigid TO that could amplify the elastic contribution in the TO response. Thus, it is important to clarify if the rotation-induced suppression of the TO response could be attributed to an unavoidable elastic effect. In our rigid TO, complicated nonlinear viscoelastic contributions are systematically eliminated. In addition, the TO operating at two different resonant frequencies allows us to decompose a possible superfluidlike frequency-independent contribution on period drop from that of the linear elastic overshoot effect. We found no substantial rotation-induced effect in the out-of-phase resonant mode unlike that found in the previous rotation experiments. It indicates that the previous rotation effect in the nonrigid TO cannot be attributed to the genuine supersolidity. According to the frequency analysis of the TO response, the frequency-dependent period drop, which can be attributed to the elastic overshoot effect, remains unaffected upon application of dc rotation. However, the frequency-independent superfluidlike contribution exhibits a strikingly different rotation effect that is currently inexplicable.