The X-ray emission from most accreting white dwarfs (WDs) in symbiotic binary stars is quite soft. Several symbiotic WDs, however, produce strong X-ray emission at energies greater than ∼20 keV. The Swift Burst Alert Telescope (BAT) instrument has detected hard X-ray emission from four such accreting WDs in symbiotic stars: RT Cru, T CrB, CD −57 3057, and CH Cyg. In one case (RT Cru), Swift detected X-rays out to greater than 50 keV at > 5σ confidence level. Combining data from the X-Ray Telescope (XRT) and BAT detectors, we find that the 0.3-150 keV spectra of RT Cru, T CrB, and CD −57 3057 are well described by emission from a single-temperature, optically thin thermal plasma, plus an unresolved 6.4-6.9 keV Fe line complex. The X-ray spectrum of CH Cyg contains an additional bright soft component. For all four systems, the spectra suffer high levels of absorption from material that both fully and partially covers the source of hard X-rays. The XRT data did not show any of the rapid, periodic variations that one would expect if the X-ray emission were due to accretion onto a rotating, highly magnetized WD. The X-rays were thus more likely from the accretion-disk boundary layer around a massive, nonmagnetic WD in each binary. The X-ray emission from RT Cru varied on timescales of a few days. This variability is consistent with being due to changes in the absorber that partially covers the source, suggesting localized absorption from a clumpy medium moving into the line of sight. The X-ray emission from CD −57 3057 and T CrB also varied during the nine months of Swift observations, in a manner that was also consistent with variable absorption.