Generalized inverses are important in statistics and other areas of applied matrix algebra. A generalized inverse of a real matrix A is a matrix H that satisfies the Moore-Penrose (M-P) property \(AHA=A\) . If H also satisfies the M-P property \(HAH=H\) , then it is called reflexive. Reflexivity of a generalized inverse is equivalent to minimum rank, a highly desirable property. We consider aspects of symmetry related to the calculation of various sparse reflexive generalized inverses of A. As is common, we use (vector) 1-norm minimization for both inducing sparsity and for keeping the magnitude of entries under control. When A is symmetric, a symmetric H is highly desirable, but generally such a restriction on H will not lead to a 1-norm minimizing reflexive generalized inverse. We investigate a block construction method to produce a symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. Letting the rank of A be r, we establish that the 1-norm minimizing generalized inverse of this type is a 1-norm minimizing symmetric generalized inverse when (i) \(r=1\) and when (ii) \(r=2\) and A is nonnegative. Another aspect of symmetry that we consider relates to another M-P property: H is ah-symmetric if AH is symmetric. The ah-symmetry property is sufficient for a generalized inverse to be used to solve the least-squares problem \(\min \{\Vert Ax-b\Vert _2:~x\in \mathbb {R}^n\}\) using H, via \(x:=Hb\) . We investigate a column block construction method to produce an ah-symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We establish that the 1-norm minimizing ah-symmetric generalized inverse of this type is a 1-norm minimizing ah-symmetric generalized inverse when (i) \(r=1\) and when (ii) \(r=2\) and A satisfies a technical condition.