  We present pulse-phase X-ray spectroscopy of the high-mass X-ray pulsar SMC
X-1 from five different epochs, using Chandra X-ray Observatory ACIS-S and
XMM-Newton EPIC-pn data. The X-ray spectrum consistently shows two distinct
components, a hard power law and a soft blackbody with kT_BB ~ 0.18 keV. For
all five epochs the hard component shows a simple double-peaked pulse shape,
and also a variation in the power law slope, which becomes harder at maximum
flux and softer at minimum flux. For the soft component, the pulse profile
changes between epochs, in both shape and phase relative to the power law
pulses. The soft component is likely produced by reprocessing of the hard X-ray
pulsar beam by the inner accretion disk. We use a model of a twisted inner
disk, illuminated by the rotating X-ray pulsar beam, to simulate pulsations in
the soft component due to this reprocessing. We find that for some disk and
beam geometries, precession of an illuminated accretion disk can roughly
reproduce the observed long-term changes in the soft pulse profiles.
