  We study the interior of distorted stationary rotating black holes on the
example of a Kerr black hole distorted by external static and axisymmetric mass
distribution. We show that there is a duality transformation between the outer
and inner horizons of the black hole, which is different from that of an
electrically charged static distorted black hole. The duality transformation is
directly related to the discrete symmetry of the space-time. The black hole
horizons area, surface gravity, and angular momentum satisfy the Smarr formula
constructed for both the horizons. We formulate the zeroth, the first, and the
second laws of black hole thermodynamics for both the horizons of the black
hole and show the correspondence between the local and the global forms of the
first law. The Smarr formula and the laws of thermodynamics formulated for both
the horizons are related by the duality transformation. The distortion is
illustrated on the example of a quadrupole and octupole fields. The distortion
fields noticeably affect the proper time of a free fall from the outer to the
inner horizon of the black hole along the symmetry semi-axes. There is some
minimal non-zero value of the quadrupole and octupole moments when the time
becomes minimal. The minimal proper time indicates the closest approach of the
horizons due to the distortion.
