One of the biologically most important parameters of the cloudy sky is the proportion P of the celestial polarization pattern available for use in animal navigation. We evaluated this parameter by measuring the polarization patterns of clear and cloudy skies using 180 degrees (full-sky) imaging polarimetry in the red (650 nm), green (550 nm) and blue (450 nm) ranges of the spectrum under clear and partly cloudy conditions. The resulting data were compared with the corresponding celestial polarization patterns calculated using the single-scattering Rayleigh model. We show convincingly that the pattern of the angle of polarization (e-vectors) in a clear sky continues underneath clouds if regions of the clouds and parts of the airspace between the clouds and the earth surface (being shady at the position of the observer) are directly lit by the sun. The scattering and polarization of direct sunlight on the cloud particles and in the air columns underneath the clouds result in the same e-vector pattern as that present in clear sky. This phenomenon can be exploited for animal navigation if the degree of polarization is higher than the perceptual threshold of the visual system, because the angle rather than the degree of polarization is the most important optical cue used in the polarization compass. Hence, the clouds reduce the extent of sky polarization pattern that is useful for animal orientation much less than has hitherto been assumed. We further demonstrate quantitatively that the shorter the wavelength, the greater the proportion of celestial polarization that can be used by animals under cloudy-sky conditions. As has already been suggested by others, this phenomenon may solve the ultraviolet paradox of polarization vision in insects such as hymenopterans and dipterans. The present study extends previous findings by using the technique of 180 degrees imaging polarimetry to measure and analyse celestial polarization patterns.