We present spectral results from Chandra and XMM-Newton observations of a sample of 22 low-redshift (z < 0:1) radio galaxies and consider whether the core emission originates from the base of a relativistic jet, or an accretion flow, or contains contributions from both. We find correlations between the unabsorbed X-ray, radio, and optical fluxes and luminosities of FR I-type radio-galaxy cores, implying a common origin in the form of a jet. On the other hand, we find that the X-ray spectra of FR II-type radio galaxy cores are dominated by absorbed emission, with N H k 10 23 atoms cm À2 , which is likely to originate in an accretion flow. We discuss several models that may account for the different nuclear properties of FR I-and FR II-type cores and also demonstrate that both heavily obscured, accretion-related and unobscured, jet-related components may be present in all radio galaxy nuclei. Any absorbed, accretion-related components in FR I-type galaxies have low radiative efficiencies.