Image analysis and computer vision can be effectively employed to recover the three-dimensional structure of imaged objects, together with their surface properties. In this paper, we address the problem of metric reconstruction and texture acquisition from a single uncalibrated view of a surface of revolution (SOR). Geometric constraints induced in the image by the symmetry properties of the SOR structure are exploited to perform self-calibration of a natural camera, 3D metric reconstruction, and texture acquisition. By exploiting the analogy with the geometry of single axis motion, we demonstrate that the imaged apparent contour and the visible segments of two imaged cross sections in a single SOR view provide enough information for these tasks. Original contributions of the paper are: single view self-calibration and reconstruction based on planar rectification, previously developed for planar surfaces, has been extended to deal also with the SOR class of curved surfaces; self-calibration is obtained by estimating both camera focal length (one parameter) and principal point (two parameters) from three independent linear constraints for the SOR fixed entities; the invariant-based description of the SOR scaling function has been extended from affine to perspective projection. The solution proposed exploits both the geometric and topological properties of the transformation that relates the apparent contour to the SOR scaling function. Therefore, with this method, a metric localization of the SOR occluded parts can be made, so as to cope with them correctly. For the reconstruction of textured SORs, texture acquisition is performed without requiring the estimation of external camera calibration parameters, but only using internal camera parameters obtained from self-calibration.