The first main result of this paper is the construction of a canonical basis for D(X). We show that it is dual to the canonical basis for P(X) that is already known.
The second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by Ardila–Postnikov, Holtz–Ron, Holtz–Ron–Xu, Li–Ron, and others. We call the underlying combinatorial structure of those spaces forward exchange matroid. A forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom.