  In this paper we present explicit formulas for the fundamental solution to
the Klein-Gordon operator on some higher dimensional generalizations of the
M\"obius strip and the Klein bottle with values in distinct pinor bundles. The
fundamental solution is described in terms of generalizations of the
Weierstra{\ss} $\wp$-function that are adapted to the context of these
geometries. The explicit formulas for the kernel then allow us to express all
solutions to the homogeneous and inhomogeneous Klein-Gordon problem with given
boundary data in the context of these manifolds. In the case of the Klein
bottle we are able to describe all null solutions of the Klein-Gordon equation
in terms of finite linear combinations of the fundamental solution and its
partial derivatives.
