The Taurus-Auriga star-forming complex hosts the only population of T Tauri stars in which an anticorrelation of X-ray activity and rotation period has been observed. We have used XMM-Newton's European Photon Imaging Cameras to perform the most sensitive survey to date of X-ray emission (0.3-10 keV) from young stars in Taurus-Auriga and investigate the dependences of X-ray activity measures -- X-ray luminosity, Lx, its ratio with the stellar luminosity, Lx/Lstar, and the surface-averaged X-ray flux, Fxs -- on rotation period. We tested for differences in the distributions of Lx/Lstar of fast and slow rotators, accretors and non-accretors, and compared the dependence of Lx/Lstar on the ratio of the rotation period and the convective turnover timescale, the Rossby number, with that of late-type main-sequence stars. We found significant anticorrelations of Lx and Fxs with rotation period, but these could be explained by the typically higher stellar luminosity and effective temperature of fast-rotators in Taurus-Auriga and a near-linear dependence of Lx on Lstar. We found no evidence for a dependence of Lx/Lstar on rotation period, but for accretors to have lower Lx/Lstar than non-accretors at all rotation periods. The Rossby numbers of accretors and non-accretors were found to be the same as those of late-type main-sequence stars showing saturated X-ray emission. We conclude that non-accreting T Tauri stars show X-ray activity entirely consistent with the saturated activity of late-type main-sequence stars. Accreting T Tauri stars show lower X-ray activity, which cannot be attributed to their slower rotation.