Excerpt: For a given linear program (LP) a permutation of its variables that sends feasible points to feasible points and preserves the objective function value of each of its feasible points is a symmetry of the LP. The set of all symmetries of an LP, denoted by GLP, is the symmetry group of the LP. Margot (2010) described a method for computing a subgroup of the symmetry group GLP of an LP. This method computes GLP when the LP has only non-redundant inequalities and its feasible set satisfies no equality constraints.
Geyer, A. J., Bulutoglu, D. A., & Ryan, K. J. (2019). Finding the symmetry group of an LP with equality constraints and its application to classifying orthogonal arrays. Discrete Optimization, 32, 93–119. https://doi.org/10.1016/j.disopt.2019.01.001