  Recently, a new type of duality was reported in some deformable mechanical
networks which exhibit Kramers-like degeneracy in phononic spectrum at the
self-dual point. In this work, we clarify the origin of this duality and
propose a design principle of 2D self-dual structures with arbitrary
complexity. We find that this duality originates from the (PCI) symmetry of the
hinge, which belongs to a more general end-fixed scaling transformation. This
symmetry gives the structure an extra degree of freedom without modifying its
dynamics. This results in , i.e., dissimilar 2D mechanical structures, either
periodic or aperiodic, having identical dynamic modes, based on which we
demonstrate a new type of wave-guide without reflection or loss. Moreover, the
PCI symmetry allows us to design various 2D periodic isostatic networks with
hinge duality. At last, by further studying a 2D non-mechanical magnonic
system, we show that the duality and the associated hidden symmetry should
exist in a broad range of Hamiltonian systems.
