Aims. The usefulness and versatility of the PSRSALSA open-source pulsar data-analysis project is demonstrated through an analysis of the radio pulsar B1839−04. This study focuses on the phenomenon of bi-drifting, an effect where the drift direction of subpulses is systematically different in different pulse profile components. Bi-drifting is extremely rare in the pulsar population, and the theoretical implications are discussed after comparing B1839−04 with the only other known bi-drifter. Methods. Various tools in PSRSALSA, including those allowing quantification of periodicities in the subpulse modulation, their flux distribution, and polarization properties, are exploited to obtain a comprehensive picture of the radio properties ...