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Generalized inverses are important in statistics and other areas of appliedmatrix algebra. A \emph{generalized inverse} of a real matrix $A$ is a matrix$H$ that satisfies the Moore-Penrose (M-P) property $AHA=A$. If $H$ alsosatisfies the M-P property $HAH=H$, then it is called \emph{reflexive}.Reflexivity of a generalized inverse is equivalent to minimum rank, a highlydesirable property. We consider aspects of symmetry related to the calculationof various \emph{sparse} reflexive generalized inverses of $A$. As is common,we use (vector) 1-norm minimization for both inducing sparsity and for keepingthe magnitude of entries under control.  When $A$ is symmetric, a symmetric $H$ is highly desirable, but generallysuch a restriction on $H$ will not lead to a 1-norm minimizing reflexivegeneralized inverse. We investigate a block construction method to produce asymmetric reflexive generalized inverse that is structured and has guaranteedsparsity. Letting the rank of $A$ be $r$, we establish that the 1-normminimizing generalized inverse of this type is a 1-norm minimizing symmetricgeneralized 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 \emph{ah-symmetric} if $AH$ is symmetric. The ah-symmetry property issufficient for a generalized inverse to be used to solve the least-squaresproblem $\min\{\|Ax-b\|_2:~x\in\mathbb{R}^n\}$ using $H$, via $x:=Hb$. Weinvestigate a column block construction method to produce an ah-symmetricreflexive generalized inverse that is structured and has guaranteed sparsity.We establish that the 1-norm minimizing ah-symmetric generalized inverse ofthis 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.

1-norm minimization and minimum-rank structured sparsity for symmetric and ah-symmetric generalized inverses: rank one and two