  We present a general scheme how to construct a substitution rule for
generating $d$-dimensional analogues of the paperfolding structures. This
substitution is proven to be primitive, so that the translation action on the
hull forms a strictly ergodic dynamical system. The substitution admits a
coincidence in the sense of Dekking, which implies that the dynamical system
has pure point spectrum. The same then holds true also for the diffraction
spectrum. The substitution also allows us to give estimates on the complexity
of the paperfolding structures, and to determine topological invariants like
the \v{C}ech cohomology groups of the hull for dimensions $d\le2$.
