Gaussian derivative filters are a good model of V1 simple cells and their action can be interpreted in the framework of differential geometry. Six such filters can together probe local structure up to 2nd order. We have factorized the 6-D space of their joint outputs by the action of a group of transformations that leave intrinsic image structure invariant. The group is generated by: spatial translation, rotation and reflection, and increasing linear-transformations of intensity. The resulting factored space is a 3-D bounded orbifold. The orbifold has non-flat intrinsic curvature, but we have found a volume-preserving, mildly distorting (mean 20%) embedding of it into Euclidean 3-space. We call the embedded orbifold the 2nd order local-image-structure solid. It is shaped like a lemon, half-flattened so that it has two creased edges running between two sharp points. The two points correspond to umbilic extremum. One crease corresponds to the varieties of pure second order structure (extrema, saddles and ridges), the other to mixtures of an umbilic extremum and a plane. The solid can be used in the study of image statistics. For example, the histogram of local 2nd order forms for natural images shows a clustering of density around effectively 1-D local forms.