  Rotating black holes exhibit a remarkable set of hidden symmetries near their
horizon. These hidden symmetries have been shown to determine phenomena such as
absorption scattering, superradiance and more recently tidal deformations, also
known as Love numbers. They have also led to a proposal for a dual thermal CFT
with left and right movers recovering the entropy of the black hole.
  In this work we provide a constructive explanation of these hidden symmetries
via analytic continuation to Klein signature. We first show that the
near-horizon region of extremal black holes is a Kleinian static solution with
mass $M$ and NUT charge $N$. We then analyze the self-dual solution, namely a
Kerr black hole with a NUT charge $N=\pm M$. Remarkably, the self-dual solution
is self-similar to its near-horizon region and hence approximate symmetries
become exact: in particular, the original two isometries of Kerr are promoted
to seven exact symmetries embedded in a conformal algebra. We analyze its full
conformal group in Kleinian twistor space, where a breaking $SO(4,2) \to
SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ occurs due to the insertion of a
preferred time direction for the black hole. Finally, we show that the spectrum
of the self-dual black hole is integrable and that the eigenvalue problem can
be mapped exactly to the Hydrogen atom where the wavefunction is solved in
terms of elementary polynomials. Perturbing to astrophysical black holes with
$N=0$, we obtain a hyperfine splitting structure.
