A model is proposed for the observed combination of power-law and thermal emission of keV X-rays from rotationally powered pulsars, For gamma-ray pulsars with accelerators very many stellar radii above the neutron star surface, 100 MeV curvature gamma-rays from e(-) or e(+) flowing starward out of such accelerators are converted to e(+/-) pairs on closed field lines all around the star. These pairs strongly affect X-ray emission from near the star in two ways. (1) The pairs are a source of synchrotron. emission immediately following their creation in regions where B similar to 10(10) G. This emission, in the photon energy range 0.1 keV less than or similar to E-X less than or similar to 5 MeV, has a power-law spectrum with energy index 0.5 and X-ray luminosity that depends on the backflow current and is typically similar to 10(33) ergs s(-1). (2) The pairs ultimately form a cyclotron resonance "blanket" surrounding the star except for two holes along the open field line bundles that pass through it. In such a blanket, the gravitational pull on e(+/-) pairs toward the star is balanced by the hugely amplified push of outflowing surface-emitted X-rays wherever cyclotron resonance occurs. Because of it, the neutron star is surrounded by a leaky "hohlraum" of hot blackbody radiation with two small holes, which prevents direct X-ray observation of a heated polar cap of a gamma-ray pulsar. Weakly spin-modulated radiation from the blanket together with more strongly spin-modulated radiation from the holes through it would then dominate observed low-energy (0.1-10 keV) emission. For non-gamma-ray pulsars, in which no such accelerators with their accompanying extreme relativistic backflow toward the star are expected, optically thick e(+/-) resonance blankets should not form (except in special cases very close to the open field line bundle). From such pulsars, blackbody radiation from both the warm stellar surface and the heated polar caps should be directly observable. In these pulsars, details of the surface magnetic field evolution, especially of polar cap areas, become relevant to observations. The models are compared to X-ray data from Geminga, PSR 1055-52, PSR 0656 + 14, PSR 1929 + 10, and PSR 0950 + 08.