  We propose an exact equivalence between the entanglement spectra of two
completely different free-fermion systems at zero temperature. This equivalence
follows from a position-momentum duality where the physical roles of the
occupied band and real space projectors are exchanged. We examine the physical
consequences of this duality in multi-band models, and as an example also
physically motivate the equivalence of the entanglement spectrum of a real
space partitioned two-band topological insulator with that of a bilayer Fermi
gas with an interlayer partition. This duality has an interesting relation with
the Wannier Spectrum in the high-temperature limit, and can also be extended to
other basis-independent physical quantities like particle-number fluctuations.
