We discuss the relation between the power carried by relativistic jets and the nuclear power provided by accretion for a group of blazars, including flat-spectrum radio quasars (FSRQs) and BL Lac objects. They are characterized by good-quality broadband X-ray data provided by the BeppoSAX satellite. The jet powers are estimated using physical parameters determined from uniformly modeling their spectral energy distributions. Our analysis indicates that for FSRQs the total jet power is of the same order as the accretion power. We suggest that blazar jets are likely powered by energy extraction from a rapidly spinning black hole (BH) through the magnetic field provided by the accretion disk. FSRQs must have large BH masses (108-109 M☉) and high, near-Eddington accretion rates. For BL Lac objects, the jet luminosity is larger than the disk luminosity. This can be understood within the same scenario if BL Lac objects have masses similar to FSRQs but accrete at largely subcritical rates, whereby the accretion disk radiates inefficiently. Thus, the ``unification'' of the two classes into a single blazar population, previously proposed on the basis of a spectral sequence governed by luminosity, finds a physical basis.