We have measured the compression-mode giant resonances—the giant monopole resonance (GMR) and the isoscalar giant dipole resonance (ISGDR—in a number of Sn (A=112-124) and Cd (A=106-116) isotopes. From the excitation energies of the GMR, we have obtained an experimental value for the asymmetry term of the nuclear incompressibility, K_tau = −550±100 MeV. Combined with the value for the incompressibility of infinite nuclear matter, K∞ = 240±20 MeV, this provides significant constraints on the interactions employed in nuclear structure and equation of state calculations. We find that the GMR centroid energies of the in both Sn and Cd isotopes are significantly lower than the theoretical predictions, pointing to the role of superfluidity in the off-shell nuclei.