  The general form of the surface stress tensor of an infinitesimally thin
shell located on a rotating null horizon is derived, when different interior
and exterior geometries are joined there. Although the induced metric on the
surface must be the same approached from either side, the first derivatives of
the metric need not be. Such discontinuities lead to a Dirac
$\delta$-distribution in the Einstein tensor localized on the horizon. For a
general stationary axisymmetric geometry the surface stress tensor can be
expressed in terms of two geometric invariants that characterize the surface,
namely the discontinuities $[\kappa]$ and $[\cal J]$ of the surface gravity
$\kappa$ and angular momentum density $\cal J$. The Komar energy and angular
momentum are given in coordinates adapted to the Killing symmetries, and the
surface contributions to each determined in terms of $[\kappa]$ and $[\cal J]$.
Guided by these, a simple modification of the original Israel junction
conditions is verified directly from the Einstein tensor density to give the
correct finite result for the surface stress, when the normal $\boldsymbol n$
to the surface is allowed to tend continuously to a null vector. The relation
to Israel's original junction conditions, which fail on null surfaces, is
given. The modified junction conditions are suitable to the matching of a
rotating `black hole' exterior to any interior geometry joined at the Kerr null
horizon surface, even when the surface normal is itself discontinuous and the
Barrab\`es-Israel formalism is also inapplicable. This joining on a rotating
null horizon is purely of the matter shell type and does not contain a
propagating gravitational shock wave.
