Several topics of the plane-wave ultra-weak variational formulations for numerically solving the Helmholtz equation are revisited: treatment of the case of piecewise constant anisotropic coefficients, derivation of the formulation, coercivity properties, etc. The construction of one of these formulations, compatible with the transmission and reflection of plane waves, at normal incidence on an interface shared by two elements of the mesh, leads to a general framework which covers all the previously used ultra-weak formulations. We thus show that any ultra-weak formulation can be characterized by its equivalence with a unique discontinuous Galerkin method for which the numerical fluxes are expressed by outgoing traces. It is also shown that the particular ultra-weak formulation, which provided the general framework, can be considered as an upwind scheme in the sense that these numerical fluxes can be obtained from a Riemann solver. Based on the theory of elliptic interface boundary-value problems, conditions on the coefficients and the geometry ensuring the coercivity properties of the formulation are brought out in the 2D case. The identification of two ways of describing plane waves in the anisotropic case and an appropriate change of variables reduce the estimates of the convergence error to those related to the usual Helmholtz equation. This also allows us to derive a theoretical basis for the choice of local plane-wave bases for an efficient coverage of the anisotropic case. Some numerical experiments illustrating the efficiency of the approach complete the study.