  We present a broadband polarization analysis of 36 discrete polarized radio
sources over a very broad, densely-sampled frequency band. Our sample was
selected on the basis of polarization behaviour apparent in narrowband archival
data at 1.4 GHz: half the sample show complicated frequency-dependent
polarization behaviour (i.e. Faraday complexity) at these frequencies, while
half show comparatively simple behaviour (i.e. they appear Faraday simple). We
re-observed the sample using the Australia Telescope Compact Array (ATCA) in
full polarization, with 6 GHz of densely sampled frequency coverage spanning
1.3 to 10 GHz. We have devised a general polarization modelling technique that
allows us to identify multiple polarized emission components in a source, and
to characterize their properties. We detect Faraday complex behaviour in almost
every source in our sample. Several sources exhibit particularly remarkable
polarization behaviour. By comparing our new and archival data, we have
identified temporal variability in the broadband integrated polarization
spectra of some sources. In a number of cases, the characteristics of the
polarized emission components, including the range of Faraday depths over which
they emit, their temporal variability, spectral index, and the linear extent of
the source, allow us to argue that the spectropolarimetric data encodes
information about the magnetoionic environment of active galactic nuclei
themselves. Furthermore, the data place direct constraints on the geometry and
magnetoionic structure of this material. We discuss the consequences of
restricted frequency bands on the detection and interpretation of polarization
structures, and implications for upcoming spectropolarimetric surveys.
