We present a reconstruction of the mean axisymmetric azimuthal and meridional flows in the Derviche Tourneur Sodium installation in Grenoble liquid sodium experiment. The experimental device sets a spherical Couette flow enclosed between two concentric spherical shells where the inner sphere holds a strong dipolar magnet, which acts as a magnetic propeller when rotated. Measurements of the mean velocity, mean induced magnetic field, and mean electric potentials have been acquired inside and outside the fluid for an inner sphere rotation rate of 9 Hz (Rm≃28). Using the induction equation to relate all measured quantities to the mean flow, we develop a nonlinear least-squares inversion procedure to reconstruct a fully coherent solution of the mean velocity field. We also include in our inversion the response of the fluid layer to the nonaxisymmetric time-dependent magnetic field that results from deviations of the imposed magnetic field from an axial dipole. The mean azimuthal velocity field we obtain shows superrotation in an inner region close to the inner sphere where the Lorentz force dominates, which contrasts with an outer geostrophic region governed by the Coriolis force, but where the magnetic torque remains the driver. The meridional circulation is strongly hindered by the presence of both the Lorentz and the Coriolis forces. Nevertheless, it contributes to a significant part of the induced magnetic energy. Our approach sets the scene for evaluating the contribution of velocity and magnetic fluctuations to the mean magnetic field, a key question for dynamo mechanisms.