  Dust polarization in millimeter (and centimeter) has been mapped in disks
around an increasing number of young stellar objects. It is usually thought to
come from emission by magnetically aligned (non-spherical) grains, but can also
be produced by dust scattering. We present a semi-analytic theory of disk
polarization that includes both the direction emission and scattering, with an
emphasis on their relative importance and how they are affected by the disk
inclination. For face-on disks, both emission and scattering tend to produce
polarization in the radial direction, making them difficult to distinguish,
although the scattering-induced polarization can switch to the azimuthal
direction if the incident radiation is beamed strongly enough in the radial
direction in the disk plane. Disk inclination affects the polarizations from
emission and scattering differently, especially on the major axis where, in the
edge-on limit, the former vanishes while the latter reaches a polarization
fraction as large as 1/3. The polarizations from the two competing mechanisms
tend to cancel each other on the major axis, producing two low polarization
"holes" (one on each side of the center) under certain conditions. We find
tantalizing evidence for at least one such "hole" in NGC1333 IRAS4A1, whose
polarization observed at 8 mm on the 100 AU scale is indicative of a pattern
dominated by scattering close to the center and by direction emission in the
outer region. If true, it would imply not only that a magnetic field exists on
the disk scale, but that it is strong enough to align large, possibly mm-sized,
grains.
