Supersolidity, the appearance of zero-viscosity flow in solids, was first indicated in helium-4 torsional oscillator (TO) experiments. In this apparatus, the irrotationality of the superfluid component causes it to decouple from the underlying normal solid, leading to a reduction in the resonant period of the TO. However, the resonant period may be altered for reasons other than supersolidity, such as the temperature dependence of the elastic modulus of solid helium. Superimposing rotation onto oscillatory measurements may distinguish between supersolidity and classical effects. We performed such simultaneous measurements of the TO and the shear modulus, and observed substantial change in the resonant period with rotational speed where the modulus remained unchanged. This contrasting behavior suggests that the decrease in the TO period is a result of supersolidity.