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, an accretion flow, or contains contributions from both. We find correlations between the unabsorbed X-ray, radio, and optical fluxes and luminosities of FRI-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 FRII-type radio-galaxy cores is dominated by absorbed emission, with $N_{rm H} > 10^{23}$ atoms cm$^{-2}$, that is likely to originate in an accretion flow. We discuss several models which may account for the different nuclear properties of FRI- and FRII-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 FRI-type galaxies have low radiative efficiencies.
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