We reconstructed the energy and the position of the shower maximum of air showers with energies $E \gtrsim 100\,$PeV using radio measurements performed with Tunka-Rex. A comparison to air-Cherenkov measurements of the same air showers with the Tunka-133 photomultiplier array confirms that the radio reconstruction works reliably. Splitting our data set into two seasons, we had blinded the Tunka-133 reconstruction for the second season, which we used as later, independent cross-check of the methods developed for the first season. This gives additional confidence in the radio reconstruction. An event-to-event comparison of Tunka-Rex and Tunka-133 shows that both experiments yield consistent values for energy and $X_{\mathrm{max}}$. The energy precision of Tunka-Rex is comparable to the Tunka-133 precision of $15\,\%$, and comes with 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 another, established method. At the moment, the $X_{\mathrm{max}}$ resolution of Tunka-Rex is approximately $40\,$g/cm$^2$. This resolution probably can be improved by deploying additional antennas and further development of the reconstruction methods, since the present analysis does not yet reveal any principle limitations.
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