Although the orientation of line segments and simple shapes is a well-studied area of vision, little is known about geometric factors that influence perceived orientation of complex multipart shapes. The study of these factors is of interest because it allows for an insight into the basic problem of how local geometric attributes are integrated perceptually into a global shape representation. We examined the perceived orientation of two-part shapes using an adjustment method and a 2AFC task. In particular, we investigated the influence of the perceptual salience, or distinctiveness, of a part--as defined by the turning angles at its boundaries--and its area relative to the main "base" part. In contrast to previous results on simple shapes, our results exhibited large and systematic deviations of perceived orientation from the principal axis of the shape. For shapes with sharp part boundaries, perceived global orientation deviated maximally from the principal axis and was approximated by the axis of the main base part of the shape. With weakening part boundaries, the perceived orientation gradually approached the principal axis of the entire shape, reflecting that both parts were taken into account in estimating orientation. The results are consistent with a differentially weighted principal-axis computation in which the attached part is given systematically lower weighting with increasing turning angles at the part boundaries. They thus allow a quantitative characterization of part salience in terms of part independence: Turning angles at a part's boundaries determine the extent to which its influence is perceptually separable from the rest of the shape. We suggest that Robust Statistics may provide a useful framework for quantifying the influence of part segmentation on visual estimation.