  In this article we consider direct and inverse problems for $\alpha$-stable,
elliptic nonlocal operators whose kernels are possibly only supported on cones
and which satisfy the structural condition of \emph{directional antilocality}
as introduced in \cite{I86}. We consider the Dirichlet problem for these
operators on the "domain of dependence of the operator" and in several, adapted
function spaces. This formulation allows one to avoid natural "gauges" which
would else have to be considered in the study of the associated inverse
problems. Exploiting the directional antilocality of these operators we
complement the investigation of the \emph{direct problem} with infinite data
and single measurement uniqueness results for the associated \emph{inverse
problems}. Here, due to the only directional antilocality, new geometric
conditions arise on the measurement domains. We discuss both the setting of
symmetric and a particular class of non-symmetric nonlocal elliptic operators,
and contrast the corresponding results for the direct and inverse problems. In
particular for only "one-sided operators" new phenomena emerge both in the
direct and inverse problems: For instance, it is possible to study the problem
in data spaces involving local and nonlocal data, the unique continuation
property may not hold in general and further restrictions on the measurement
set for the inverse problem arise.
