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 [CII], 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 an accuracy of within 5% for most of the EoR, reaching 0.6% accuracy on a scale of $ksim0.1,text{Mpc}^{-1}$ at $left< x_i right> = 0.36$ ($z = 8.34$ 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.
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