  We reconstructed the energy and the position of the shower maximum of air
showers with energies $E \gtrsim 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 $X_{\mathrm{max}}$ 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 $X_{\mathrm{max}}$, this is the first direct experimental
correlation of radio measurements with a different, established method. At the
moment, the $X_{\mathrm{max}}$ resolution of Tunka-Rex is approximately $40
$g/cm$^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.
