The large-scale structure of the Universe should soon be measured at high redshift during the Epoch of Reionization ( EoR ) through line-intensity mapping . A number of ongoing and planned surveys are using the 21 cm line to trace neutral hydrogen fluctuations in the intergalactic medium ( IGM ) during the EoR . These may be fruitfully combined with separate efforts to measure large-scale emission fluctuations from galactic lines such as [ C ii ] , CO , H- \alpha , and Ly- \alpha during the same epoch . The large scale power spectrum of each line encodes important information about reionization , with the 21 cm power spectrum providing a relatively direct tracer of the ionization history . Here we show that the large scale 21 cm power spectrum can be extracted using only cross-power spectra between the 21 cm fluctuations and each of two separate line-intensity mapping data cubes . This technique is more robust to residual foregrounds than the usual 21 cm auto-power spectrum measurements and so can help in verifying auto-spectrum detections . We characterize the accuracy of this method using numerical simulations and find that the large-scale 21 cm power spectrum can be inferred to a simulated accuracy of within 5 \% for most of the EoR . Our estimate of the 21 cm power spectrum reaches 0.6 \% accuracy on a scale of k \sim 0.1 \text { Mpc } ^ { -1 } at \langle x _ { i } \rangle = 0.36 ( z = 8.34 in our model ) . An extension from two to N additional lines would provide N ( N - 1 ) / 2 cross-checks on the large-scale 21 cm power spectrum . This work strongly motivates redundant line-intensity mapping surveys probing the same cosmological volumes .