We develop a new method of interpreting large cardinal axioms as giving rise to topological symmetries of the universe of sets, similar to the construction of Fraenkel-Mostowski-Specker models. This allows us to define a "symmetric" inner model construction. In this vein, we use Fraenkel-Mostowski-Specker type stabiliser arguments to deduce the Kunen inconsistency, as well as resolving an open problem of Rogers.