Extragalactic radio sources produce radio waves and narrow emission lines by very different physical processes: synchrotron radio emission arises from lobes filled with magnetized plasma extending over scales of kiloparsecs to megaparsecs, fed by the total kinetic power, Q, of jets driven by a central engine, whereas narrow-line luminosity L(NLR) arises from gas, typically concentrated in the inner few kiloparsecs, that has been photoionized by a nuclear source. We report here the discovery of a close relationship between Q and L(NLR) - an approximate proportionality which extends over four orders of magnitude from low-Q radio sources with relaxed structures to high-Q, radio-luminous classical double-lobe radio galaxies. Objects with broad Balmer lines follow the same trend as those without, showing that quasar-like photoionizing sources are ubiquitous but not always obvious. Moreover, all radio-source central engines channel at least as much power into the jets as is radiated by accretion: this high efficiency implies that the engine is a massive spinning black hole which both powers the jets and controls the accretion rate.