This paper considers the problem of 3D reconstruction from 2D points in one or more images and auxiliary information about the corresponding 3D features: alignments, coplanarities, ratios of lengths or symmetries are known. Our first contribution is a necessary and sufficient criterion that indicates whether a dataset, or subsets thereof, defines a rigid reconstruction up to scale and translation. Another contribution is a reconstruction method for one or more images. We show that the observations impose linear constraints on the reconstruction. All the input data, possibly coming from many images, is summarized in a single linear system, whose solution yields the reconstruction. The criterion which indicates whether the solution is unique up to scale and translation is the rank of another linear system, called the “twin” system. Multiple objects whose relative scale can be arbitrarily chosen are identified. The reconstruction is obtained up to an affine transformation, or, if calibration is available, up to a Euclidean transformation.