Thomson backscattering of intense laser pulses from relativistic electrons not only allows for the generation of bright x-ray pulses but also for the investigation of the complex particle dynamics at the interaction point. For this purpose a complete spectral characterization of a Thomson source powered by a compact linear electron accelerator is performed with unprecedented angular and energy resolution. A rigorous statistical analysis comparing experimental data to 3D simulations enables, e.g., the extraction of the angular distribution of electrons with 1.5% accuracy and, in total, provides predictive capability for the future high brightness hard x-ray source PHOENIX (photon electron collider for narrow bandwidth intense x rays) and potential gamma-ray sources.