  In this paper we study the self-adjointness and spectral properties of
two-dimensional Dirac operators with electrostatic, Lorentz scalar, and
anomalous magnetic $\delta$-shell interactions with constant weights that are
supported on a smooth unbounded curve that is straight outside a compact set
and whose ends are rays that are not parallel to each other. For all possible
combinations of interaction strengths we describe the self-adjoint realizations
and compute their essential spectra. Moreover, we prove in different situations
the existence of geometrically induced discrete eigenvalues.
