  The photoionization models for the narrow emission line regions of powerful
radio galaxies have yet to be tested in depth. To this end, we present
high-quality long-slit spectroscopy of the powerful double-nucleus radio galaxy
3C 321. The data have good enough spatial resolution to be able to trace the
variation in emission-line properties on kpc scales. Continuum modelling and
subtraction enables the faint emission line fluxes to be measured in several
regions across the emission line nebula. We plot diagnostic line-ratio diagrams
and compare them with the predictions of various photoionization models,
finding that the data is best fit by models which assume a mixture of optically
thin and thick clouds illuminated by a power-law continuum.
  The emission line kinematics, line ratios and deduced physical conditions
show remarkably little variation across the source. We deduce a mean electron
density of 400 +/- 120 cm-3 and a mean temperature of 11500 +/- 1500 K.
Assuming a single population of optically thick line-emitting clouds, we
calculate a mean photoionization parameter of (1.1 +/- 0.5) x 10e-2 and hence a
photoionizing photon luminosity of Q ~ 10e55 -- 10e56 photon/s/sr. This
indicates a central engine as luminous as that of the powerful quasar 3C 273,
yet there is no evidence for such an energetically prolific central engine at
either far-infrared or radio wavelengths. We therefore conclude that the
mixed-media models, which give Q ~ 5 x 10e53 -- 5 x 10e54, represent a more
likely scenario.
  As a by-product of the continuum subtraction we infer that young stellar
populations account for ~ 0.4% of the visible stellar mass in the galaxy, and
that these populations are spatially extended.
