We analyze Hubble Space Telescope observations of scattering regions in 20 luminous obscured quasars at $0.24<z<0.65$ (11 new observations and 9 archival ones) observed at rest-frame $sim 3000$AA. We find spectacular $5-10$ kpc-scale scattering regions in almost all cases. The median scattering efficiency at this wavelength (the ratio of observed to estimated intrinsic flux) is 2.3%, and 73% of the observed flux at this wavelength is due to scattered light, which if unaccounted for may strongly bias estimates of quasar hosts star formation rates. Modeling these regions as illuminated dusty cones, we estimate the radial density distributions of the interstellar medium as well as the geometric properties of circumnuclear quasar obscuration -- inclinations and covering factors. Small derived opening angles (median half-angle and standard deviation 27dg$pm$9dg) are inconsistent with a 1:1 type 1 / type 2 ratio. We suggest that quasar obscuration is patchy and that the observer has a $sim 40%$ chance of seeing a type 1 source even through the obscuration. We estimate median density profile of the scattering medium to be $n_{rm H}=0.04-0.5$ $(1{rm kpc}/r)^2$ cm$^{-3}$, depending on the method. Quasars in our sample likely exhibit galaxy-wide winds, but if these consist of optically thick clouds then only a small fraction of the wind mass ($la 10%$) contributes to scattering.
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