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 behavior apparent in narrowband archival data at 1.4GHz: half the sample shows complicated frequency-dependent polarization behavior (i.e., Faraday complexity) at these frequencies, while half shows comparatively simple behavior (i.e., they appear Faraday simple). We re-observed the sample using the Australia Telescope Compact Array in full polarization, with 6GHz of densely sampled frequency coverage spanning 1.3-10GHz. We have devised a general polarization modeling technique that allows us to identify multiple polarized emission components in a source, and to characterize their properties. We detect Faraday complex behavior in almost every source in our sample. Several sources exhibit particularly remarkable polarization behavior. 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 encode information about the magneto-ionic environment of active galactic nuclei themselves. Furthermore, the data place direct constraints on the geometry and magneto-ionic structure of this material. We discuss the consequences of restricted frequency bands on the detection and interpretation of polarization structures, and the implications for upcoming spectropolarimetric surveys.