© 2019. The American Astronomical Society. All rights reserved. Foreground power dominates the measurements of interferometers that seek a statistical detection of highly-redshifted H i emission from the Epoch of Reionization (EoR). The chromaticity of the instrument creates a boundary in the Fourier transform of frequency (proportional to k ∥) between spectrally smooth emission, characteristic of the strong synchrotron foreground (the "wedge"), and the spectrally structured emission from H i in the EoR (the "EoR window"). Faraday rotation can inject spectral structure into otherwise smooth polarized foreground emission, which through instrument effects or miscalibration could possibly pollute the EoR window. For instruments pursuing a "foreground avoidance" strategy of simply measuring in the EoR window, and not attempting to model and remove foregrounds, as is the plan for the first stage of the Hydrogen Epoch of Reionization Array (HERA), characterizing the intrinsic instrument polarization response is particularly important. Using data from the HERA 19-element commissioning array, we investigate the polarization response of this new instrument in the power-spectrum domain. We perform a simple image-based calibration based on the unpolarized diffuse emission of the Global Sky Model, and show that it achieves qualitative redundancy between the nominally redundant baselines of the array and reasonable amplitude accuracy. We construct power spectra of all fully polarized coherencies in all pseudo-Stokes parameters, and discuss the achieved isolation of foreground power due to the intrinsic spectral smoothness of the foregrounds, the instrument chromaticity, and the calibration. We compare to simulations based on an unpolarized diffuse sky model and detailed electromagnetic simulations of the dish and feed, confirming that in Stokes I, the calibration does not add significant spectral structure beyond that expected from the interferometer array configuration and the modeled primary beam response. Furthermore, this calibration is stable over the 8 days of observations considered. Excess power is seen in the power spectra of the linear polarization Stokes parameters, which is not easily attributable to leakage via the primary beam, and results from some combination of residual calibration errors and actual polarized emission. Stokes V is found to be highly discrepant from the expectation of zero power, strongly pointing to the need for more accurate polarized calibration.