  We introduce the notions of directional dynamical cubes and directional
regionally proximal relation defined via these cubes for a minimal
$\mathbb{Z}^d$-system $(X,T_1,\ldots,T_d)$. We study the structural properties
of systems that satisfy the so called unique closing parallelepiped property
and we characterize them in several ways. In the distal case, we build the
maximal factor of a $\mathbb{Z}^d$-system $(X,T_1,\ldots,T_d)$ that satisfies
this property by taking the quotient with respect to the directional regionally
proximal relation. Finally, we completely describe distal
$\mathbb{Z}^d$-systems that enjoy the unique closing parallelepiped property
and provide explicit examples.
