Wolf-Rayet stars have strong, hot winds, with mass-loss rates at least a factor of ten greater than their O-star progenitors, although their terminal wind speeds are similar. In this paper we use the technique of multiband linear polarimetry to extract information on the global asymmetry of the wind in a sample of 47 bright Galactic WR stars. Our observations also include time-dependent observations of 17 stars in the sample. The path to our goal includes removing the dominating component of wavelength-dependent interstellar polarization (ISP), which normally follows the well-known Serkowski law. We include a wavelength-dependent ISP position angle parameter in our ISP law and find that 15 stars show significant results for this parameter. We detect a significant component of wavelength-independent polarization due to electron scattering in the wind for 10 cases, with most WR stars showing none at the $sim$0.05% level precision of our data. The intrinsically polarized stars can be explained with binary interaction, large-scale wind structure, and clumping. We also found that 5 stars out of 19 observed with the Stromgren $b$ filter (probing the complex $lambda$4600--4700 emission line region) have significant residuals from the ISP law and propose that this is due to binary illumination or wind clumping. We provide a useful catalogue of ISP for 47 bright Galactic WR stars and upper limits on the possible level of intrinsic polarization.
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