Previous studies in low pressure magnetized capacitively coupled radio frequency (RF) plasmas operated in argon with optimized geometric reactor symmetry have shown that the magnetic asymmetry effect (MAE) allows to control the particle flux energy distributions at the electrodes, the plasma symmetry, and the DC self-bias voltage by tuning the magnetron-like magnetic field adjacent to one electrode (Oberberg et al 2019 Plasma Sources Sci. Technol. 28 115021; Oberberg et al 2018 Plasma Sources Sci. Technol. 27 105018). In this way non-linear electron resonance heating (NERH) induced via the self-excitation of the plasma series resonance (PSR) was also found to be controllable. Such plasma sources are frequently used for reactive RF magnetron sputtering, but the discharge conditions used for such applications are significantly different compared to those studied previously. A high DC self-bias voltage (generated via a geometric reactor asymmetry) is required to realize a sufficiently high ion bombardment energy at the target electrode and a reactive gas must be added to deposit ceramic compound layers. Thus in this work, the MAE is investigated experimentally in a geometrically asymmetric capacitively coupled RF discharge driven at 13.56 MHz and operated in mixtures of argon and oxygen. The DC self-bias, the symmetry parameter, the time resolved RF current, the plasma density, and the mean ion energy at the grounded electrode are measured as a function of the driving voltage amplitude and the magnetic field at the powered electrode. Results obtained in pure argon discharges are compared to measurements performed in argon with reactive gas admixture. The results reveal a dominance of the geometrical over the magnetic asymmetry. The DC self-bias voltage as well as the symmetry parameter are found to be only weakly influenced by a change of the magnetic field compared to previous results obtained in a geometrically more symmetric reactor. Nevertheless, the magnetic field is found to provide the opportunity to control NERH magnetically also in geometrically asymmetric reactors. Adding oxygen does not alter these discharge properties significantly compared to a pure argon discharge.