In [R1] a notion of restricted orbit equivalence for ergodictransformations was developed. Here we modify that structure in orderto generalize it to actions of higher-dimensional groups, in particular${\Bbb Z}^d$-actions. The concept of a ‘size’ is developed first from anaxiomatized notion of the size of a permutation of a finite blockin ${\Bbb Z}^d$. This is extended to orbit equivalences which arecohomologousto the identity and, via the natural completion, to a notion ofrestrictedorbit equivalence. This is shown to be an equivalence relation.Associated to each size is an entropy which is an equivalence invariant.As in the one-dimensional case this entropy is either the classicalentropyor is zero. Several examples are discussed.