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The high-redshift 21 cm signal from the Epoch of Reionization (EoR) is a promising observational probe of the early universe. Current- and next-generation radio interferometers such as the Hydrogen Epoch of Reionization Array (HERA) and Square Kilometre Array (SKA) are projected to measure the 21 cm auto power spectrum from the EoR. Another observational signal of this era is the kinetic Sunyaev-Zel'dovich (kSZ) signal in the cosmic microwave background (CMB), which will be observed by the upcoming Simons Observatory (SO) and CMB-S4 experiments. The 21 cm signal and the contribution to the kSZ from the EoR are expected to be anti-correlated, the former coming from regions of neutral gas in the intergalactic medium and the latter coming from ionized regions. However, the naive cross-correlation between the kSZ and 21 cm maps suffers from a cancellation that occurs because ionized regions are equally likely to be moving toward or away from the observer and so there is no net correlation with the 21 cm signal. We present here an investigation of the 21 cm-kSZ-kSZ bispectrum, which should not suffer the same cancellation as the simple two-point cross-correlation. We show that there is a significant and non-vanishing signal that is sensitive to the reionization history, suggesting the statistic may be used to confirm or infer the ionization fraction as a function of redshift. In the absence of foreground contamination, we forecast that this signal is detectable at high statistical significance with HERA and SO. The bispectrum we study suffers from the fact that the kSZ signal is sensitive only to Fourier modes with long-wavelength line-of-sight components, which are generally lost in the 21 cm data sets owing to foreground contamination. We discuss possible strategies for alleviating this contamination, including an alternative four-point statistic that may help circumvent this issue.