We present new ROSAT PSPC X-ray pointed observations of nine Abell clusters that contain wide-angle tailed (WAT) radio galaxies. The sample of WAT clusters was selected from a VLA radio survey of the Abell catalog at 20 cm and the ROSAT All-Sky Survey (RASS). We found that up to eight (89%) clusters show some evidence of X-ray substructure. These clusters also have clear evidence of isophotal twisting, ellipticity variation, and isophote centroid variation as a function of distance from the central X-ray peak. Moreover, none of these WAT clusters have significant cluster-wide cooling flows in spite of the presence of dominant elliptical (radio) galaxies. We performed temperature fits to the X-ray emission by applying a Raymond-Smith fit to the spectra and a ``broadband'' temperature technique. One striking correlation with the radio sources was discovered: there is a strong relationship between the orientation of the WAT tails and the direction of X-ray elongations within the core of the clusters. We believe that this is consistent with WAT clusters undergoing mergers with groups or subclusters of galaxies. We compared our results with numerical hydro/N-body simulations of merging galaxy clusters to test this hypothesis. We found that the X-ray synthetic images generated by the simulations reproduce all the major morphological properties that we found in our sample. Furthermore, the simulations also show that the bulk flow of gas within the ICM present during the merger could be responsible for the correlation between the X-ray elongations and the direction of the bending of the radio tails.