In [R1] a notion of restricted orbit equivalence for ergodic transformations was developed. Here we modify that structure in order to generalize it to actions of higher-dimensional groups, in particular ${\Bbb Z}^d$-actions. The concept of a ‘size’ is developed first from an axiomatized notion of the size of a permutation of a finite block in ${\Bbb Z}^d$. This is extended to orbit equivalences which are cohomologous to the identity and, via the natural completion, to a notion of restricted orbit 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 classical entropy or is zero. Several examples are discussed.