We reconstructed the energy and the position of the shower maximum of air showers with energies E ∼> 100 PeV applying a method using radio measurements performed with Tunka-Rex. An event-to-event comparison to air-Cherenkov measurements of the same air showers with the Tunka-133 photomultiplier array confirms that the radio reconstruction works reliably. The Tunka-Rex reconstruction methods and absolute scales have been tuned on CoREAS simulations and yield energy and X{sub max} values consistent with the Tunka-133 measurements. The results of two independent measurement seasons agree within statistical uncertainties, which gives additional confidence in the radio reconstruction. The energy precision of Tunka-Rex is comparable to the Tunka-133 precision of 15%, and exhibits a 20% uncertainty on the absolute scale dominated by the amplitude calibration of the antennas. For X{sub max}, this is the first direct experimental correlation of radio measurements with a different, established method. At the moment, the X{sub max} resolution of Tunka-Rex is approximately 40 g/cm{sup 2}. This resolution can probably be improved by deploying additional antennas and by further development of the reconstruction methods, since the present analysis does not yet reveal any principle limitations.