Context. Galactic dust emission is polarized at unexpectedly high levels, as revealed by Planck. Aims. The origin of the observed ≃20% polarization fractions can be identified by characterizing the properties of optical starlight polarization in a region with maximally polarized dust emission. Methods. We measure the R-band linear polarization of 22 stars in a region with a submillimeter polarization fraction of ≃20%. A subset of 6 stars is also measured in the B, V, and I bands to investigate the wavelength dependence of polarization. Results. We find that starlight is polarized at correspondingly high levels. Through multiband polarimetry we find that the high polarization fractions are unlikely to arise from unusual dust properties, such as enhanced grain alignment. Instead, a favorable magnetic field geometry is the most likely explanation, and is supported by observational probes of the magnetic field morphology. The observed starlight polarization exceeds the classical upper limit of [p_V/E(B−V)]_max = 9% mag⁻¹ and is at least as high as 13% mag⁻¹, as inferred from a joint analysis of Planck data, starlight polarization, and reddening measurements. Thus, we confirm that the intrinsic polarizing ability of dust grains at optical wavelengths has long been underestimated.