We investigate the discharge characteristics of a low-pressure geometrically asymmetric cylindrical capacitively coupled plasma discharge with an axisymmetric magnetic field generating an E × B drift in the azimuthal direction. Vital discharge parameters, including electron density, electron temperature, DC self-bias, and electron energy probability function (EEPF), are studied experimentally for different magnetic field strength (B) values. A transition in the plasma parameters is observed for a specific range of magnetic fields where the discharge is highly efficient with lower electron temperature. Outside this range of magnetic field, the plasma density drops, followed by an increase in the electron temperature. The observed behavior is attributed to the transition from geometrical asymmetry to magnetic field-associated symmetry due to reduced radial losses and plasma confinement in the peripheral region. The DC self-bias increases almost linearly from a large negative value to nearly zero, i.e., it turns into a symmetric discharge. The EEPF undergoes a transition from bi-Maxwellian for unmagnetized to Maxwellian at intermediate B and finally becomes a weakly bi-Maxwellian at higher values of B. The above transitions present a novel way to independently control the ion energy and ion flux in a cylindrical capacitively coupled plasma system using an axisymmetric magnetic field with an enhanced plasma density and lower electron temperature that is beneficial for plasma processing applications.