The differential cross section for Compton scattering d2σ/dΩdE in germanium (Z=32) was measured using the sensitive volume of a Ge detector as the scatterer and another Ge detector for detection of the scattered radiation, at an incident energy of 59.537 keV and a scattering angle of about 170°. The application of the coincidence technique and the requirement of a constant energy sum yield a clear spectrum in a broad energy range. Detailed analyses of the processes involved in the detector-to-detector scattering were made, including various double-scattering processes. Calculations show that bremsstrahlung of photoelectrons dominates at low energies, while all double-scattering processes weakly contribute to the coincidence rates. The influence of Compton-Rayleigh and Rayleigh-Compton scattering on Compton data is relatively stronger at high energy, while an approximate proportionality of Compton-Compton and single Compton spectra at scattering angles close to 180° was obtained, assuming the impulse approximation. Single and double Compton scattering on stationary electrons at 180° have been shown to produce photons of exactly the same energy. The measured differential cross sections for Compton scattering and results of calculations based on the impulse approximation are in fair agreement.