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We present high-resolution (∼30 au) ALMA Band 6 dust polarization observations of VLA 1623. The VLA 1623 data resolve compact ∼40 au inner disks around the two protobinary sources, VLA 1623-A and VLA 1623-B, and also an extended ∼180 au ring of dust around VLA 1623-A. This dust ring was previously identified as a large disk in lower-resolution observations. We detect highly structured dust polarization toward the inner disks and the extended ring with typical polarization fractions ≃1.7% and ≃2.4%, respectively. The two components also show distinct polarization morphologies. The inner disks have uniform polarization angles aligned with their minor axes. This morphology is consistent with expectations from dust scattering. By contrast, the extended dust ring has an azimuthal polarization morphology not previously seen in lower-resolution observations. We find that our observations are well-fit by a static, oblate spheroid model with a flux-frozen, poloidal magnetic field. We propose that the polarization traces magnetic grain alignment likely from flux freezing on large scales and magnetic diffusion on small scales. Alternatively, the azimuthal polarization may be attributed to grain alignment by the anisotropic radiation field. If the grains are radiatively aligned, then our observations indicate that large (∼100 µm) dust grains grow quickly at large angular extents. Finally, we identify significant proper motion of VLA 1623 using our observations and those in the literature. This result indicates that the proper motion of nearby systems must be corrected for when combining ALMA data from different epochs.