Quasars with similar core-dominant radio properties can be classified by their differences at optical and infrared frequencies. Their X-ray properties, however, might be expected to be similar if, as is commonly believed, the synchrotron self-Compton mechanism relates their radio and X-ray emission. We have compared the 0.1-3.5 keV mean power-law energy spectral indices, ᾱ, as measured with the Einstein Observatory IPC, for three classes: (i) 19 low-polarization flat radio spectrum, core-dominant QSOs (FRS QSOs), (ii) 12 highly polarized QSOs (HPQs), and (iii) 24 radio-selected BL Lac objects. A power-law spectrum with energy index α gives an acceptable fit to the data for each source. We present individual spectral fits for a few sources for which results are not presented elsewhere. The likelihood distribution for α was computed for each source with the normalization and column density of intervening Galactic matter as free parameters. A maximum-likelihood analysis was used to find the mean power-law index, ᾱ, and the standard deviation, σ, for each class, assuming the intrinsic distribution to be Gaussian. Results for the HPQs and FRS QSOs are similar, with a best fit of ᾱ ≈ 0.5, whereas for the BL Lac objects, ᾱ ≈ 1.0. Each of the three samples is consistent with all its objects emitting a spectrum with the same spectral index. This is in contrast to a small sample of X-ray-selected BL Lac objects, which also fit ᾱ ≈ 1.0, where IPC observations are inconsistent with a single spectral index. The X-ray spectral difference between HPQs and radio-selected BL Lac objects questions the wisdom of treating these objects together in statistical analyses, and it is a possible difficulty for the hypothesis that BL Lac objects are gravitationally lensed HPQs. We address the implications of our X-ray results for synchrotron self-Compton models.