Self-piercing riveting (SPR) is a mature method to join dissimilar materials in vehicle body assembling. Friction self-piercing riveting (F-SPR) is a newly developed technology for joining low-ductility materials by combining SPR and friction stir spot welding processes. In this paper, the SPR and F-SPR were employed to join AA6061-T6 aluminum alloy and AZ31B magnesium alloy. The two processes were studied in parallel to investigate the effects of stack orientation on riveting force, macro-geometrical features, hardness distributions, and mechanical performance of the joints. The results indicate that both processes exhibit a better overall joint quality by riveting from AZ31B to AA6061-T6. Major cracking in the Mg sheet is produced when riveting from AA6061-T6 to AZ31B in the case of SPR, and the cracking is inhibited with the thermal softening effect by friction heat in the case of F-SPR. The F-SPR process requires approximately one-third of the riveting forces of the SPR process but exhibits a maximum of 45.4% and 59.1% higher tensile–shear strength for the stack orientation with AZ31B on top of AA6061-T6 and the opposite direction, respectively, than those of the SPR joints. The stack orientation of riveting from AZ31B to AA6061-T6 renders better cross-section quality and higher tensile–shear strength and is recommended for both processes.