We study a mathematical framework for the construction of domain decomposition methods (DDM) for the model Helmholtz equation with high order transmission conditions (TC) and cross-points. A compatibility condition is formulated for cross-points matrices so that the DDM is proved to be convergent under general conditions. The proof is based on a new global energy formulated on the skeleton of the DDM decomposition. It allows to use transmission operators which have a non trivial anti-symmetric part.