We have measured mid-infrared radiation from an orientation-unbiased sample of 3CRR galaxies and quasars at redshifts 0.4 < z < 1.2 with the IRS and MIPS instruments on the Spitzer Space Telescope. Powerful emission (L_24micron > 10^22.4 W/Hz/sr) was detected from all but one of the sources. We fit the Spitzer data as well as other measurements from the literature with synchrotron and dust components. The IRS data provide powerful constraints on the fits. At 15 microns, quasars are typically four times brighter than radio galaxies with the same isotropic radio power. Based on our fits, half of this difference can be attributed to the presence of non-thermal emission in the quasars but not the radio galaxies. The other half is consistent with dust absorption in the radio galaxies but not the quasars. Fitted optical depths are anti-correlated with core dominance, from which we infer an equatorial distribution of dust around the central engine. The median optical depth at 9.7 microns for objects with core-dominance factor R > 10^-2 is approximately 0.4; for objects with R < 10^-2, it is 1.1. We have thus addressed a long-standing question in the unification of FR II quasars and galaxies: quasars are more luminous in the mid-infrared than galaxies because of a combination of Doppler-boosted synchrotron emission in quasars and extinction in galaxies, both orientation-dependent effects.