I think there are at least three distinct views on what to do about symmetry in MIP models: eliminate it from the model (e.g., Sherali & Smith 2001); eliminate it via cuts during branch-and-cut (e.g., Margot 2002); or exploit it during branching (e.g., Ostrowski, Linderoth, Rossi & Smriglio 2011). The constraint programming folks are also painfully aware of symmetry, and have their own ways of dealing with it. I’m not sure if those methods map to the aforementioned.
As a math major, I find the issue a trifle ironic. In all other areas of mathematics, symmetry is the mathematician’s friend.

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