The best prospects for robust, extensible Natural Language Understanding systems will come through the use of a semantics with a sound fit to way language is organized into units. Many single words correspond to relations rather than terms; many full phrases (maximal projections) correspond to lambda abstractions rather than wffs. In this paper we explore the consequences of some of these common but syntactically unconventional phrase types on the representations that could be used to represent their meanings. These phrases----conjunction reductions without semantically natural heads, and displaced constituents that must be given interim representations until the parse reaches a point where they can be completed--are common in real texts, and must be accounted for in any successful language understanding system. We use them to illustrate some of the unusual properties of the representational system we work with, including first class representations of the links between relations and their arguments and first class representations of partially saturated relations.