We present a highly tractable non-singular modification of the Kerr geometry,
dubbed the "eye of the storm" -- a rotating regular black hole with an
asymptotically Minkowski core. This is achieved by "exponentially suppressing"
the mass parameter in the Kerr spacetime: $m\to m \; e^{-\ell/r}$. The single
parameter $\ell$ quantifies the deviation from the usual Kerr spacetime. Some
of the classical energy conditions are globally satisfied, whilst certain
choices for $\ell$ force any energy-condition-violating physics into the deep
core. The geometry possesses the full "Killing tower'" of principal tensor,
Killing--Yano tensor, and nontrivial Killing tensor, with associated Carter
constant; hence the Hamilton--Jacobi equations are separable, and the geodesics
integrable. The Klein--Gordon equation is also separable on this candidate
spacetime. The tightly controlled deviation from Kerr renders the physics
extraordinarily tractable when compared with analogous results in the current
literature. This spacetime will be amenable to straightforward extraction of
astrophysical observables falsifiable/ verifiable by the experimental
community.