We explore the descriptive power, in terms of syntactic phenomena, of a formalism that extends TreeAdjoining Grammar (TAG) by adding a fourth level of hierarchical decomposition to the three levels TAG already employs. While extending the descriptive power minimally, the additional level of decomposition allows us to obtain a uniform account of a range of phenomena that has heretofore been difficult to encompass, an account that employs unitary elementary structures and eschews synchronized derivation operations, and which is, in many respects, closer to the spirit of the intuitions underlying TAG-based linguistic theory than previously considered extensions to TAG.