  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.
