  We consider the analytic continuation of $(p+q)$-dimensional Minkowski space
(with $p$ and $q$ even) to $(p,q)$-signature, and study the conformal boundary
of the resulting "Klein space". Unlike the familiar $(-+++..)$ signature, now
the null infinity ${\mathcal I}$ has only one connected component. The spatial
and timelike infinities ($i^0$ and $i'$) are quotients of generalizations of
AdS spaces to non-standard signature. Together, ${\mathcal I}, i^0$ and $i'$
combine to produce the topological boundary $S^{p+q-1}$ as an $S^{p-1} \times
S^{q-1}$ fibration over a null segment. The highest weight states (the
$L$-primaries) and descendants of $SO(p,q)$ with integral weights give rise to
natural scattering states. One can also define $H$-primaries which are highest
weight with respect to a signature-mixing version of the Cartan-Weyl generators
that leave a point on the celestial $S^{p-1} \times S^{q-1}$ fixed. These
correspond to massless particles that emerge at that point and are Mellin
transforms of plane wave states.
