We present new BeppoSAX LECS and MECS observations, covering the energy range 0.1 - 10 keV (observers frame), of four BL Lacertae objects selected from the 1 Jy sample. All sources display a flat (alpha_x ~ 0.7) X-ray spectrum, which we interpret as inverse Compton emission. One object shows evidence for a low-energy steepening (Delta alpha_x ~ 0.9) which is likely due to the synchrotron component merging into the inverse Compton one around ~ 2 keV. A variable synchrotron tail would explain why the ROSAT spectra of our sources are typically steeper than the BeppoSAX ones (Delta alpha_x} ~ 0.7). The broad-band spectral energy distributions fully confirm this picture and model fits using a synchrotron inverse Compton model allow us to derive the physical parameters (intrinsic power, magnetic field, etc.) of our sources. By combining the results of this paper with those previously obtained on other sources we present a detailed study of the BeppoSAX properties of a well-defined sub-sample of 14 X-ray bright (f_x (0.1 - 10 keV) > 3 x 10^{-12} erg/cm^2/s) 1-Jy BL Lacs. We find a very tight proportionality between nearly simultaneous radio and X-ray powers for the 1-Jy sources in which the X-ray band is dominated by inverse Compton emission, which points to a strong link between X-ray and radio emission components in these objects.
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