  Let $m\neq n$. An $m\times n\times p$ proper array is a three-dimensional
array composed of directed cubes that obeys certain constraints. Because of
these constraints, the $m\times n\times p$ proper arrays may be classified via
a schema in which each $m\times n\times p$ proper array is associated with a
particular $m\times n$ planar face. By representing each connencted component
present in the $m\times n$ planar face with a distinct letter, and the position
of each outward pointing connector by a circle, an $m\times n$ array of circled
letters is formed. This $m\times n$ array of circled letters is the word
representation associated with the $m\times n\times p$ proper array. The main
result of this paper involves the enumeration of all $m\times n$ word
representations modulo symmetry, where the symmetry is derived from the group
$D_2 = C_2\times C_2$ acting on the set of word representations. This
enumeration is achieved by forming a linear combination of four exponential
generating functions, each of which is derived from a particular symmetry
operation. This linear combination counts the number of partitions of the set
of $m\times n$ words representations that are inequivalent under $D_2$.
