In a system with chiral topological order, there is a remarkable correspondence between the edge and entanglement spectra: the low-energy spectrum of the system in the presence of a physical edge coincides with the lowest part of the entanglement spectrum (ES) across a virtual cut of the system into two parts, up to rescaling and shifting. This correspondence is believed to be due to the existence of protected gapless edge modes. In this paper, we explore whether the edge-entanglement spectrum correspondence extends to nonchiral topological phases, where there are no protected gapless edge modes. Specifically, we consider the Wen-plaquette model, which is equivalent to the Kitaev toric code model and has Z2 topological order (quantum double of Z2). The unperturbed Wen-plaquette model displays an exact correspondence: both the edge and entanglement spectra within each topological sector a(a=1,⋯,4) are flat and equally degenerate. Here, we show, through a detailed microscopic calculation, that in the presence of generic local perturbations: (i) the effective degrees of freedom for both the physical edge and the entanglement cut consist of a (spin-1/2) spin chain, with effective Hamiltonians Hedgea and Henta, respectively, both of which have a Z2 symmetry enforced by the bulk topological order; (ii) there is in general no match between the low-energy spectra of Hedgea and Henta, that is, there is no edge-ES correspondence. However, if supplement the Z2 topological order with a global symmetry (translational invariance along the edge/entanglement cut), i.e., by considering the Wen-plaquette model as a symmetry-enriched topological phase (SET), then there is a finite domain in Hamiltonian space in which both Hedgea and Henta realize the critical Ising model, whose low-energy effective theory is the c=1/2 Ising CFT. This is achieved because the presence of the global symmetry implies that the effective degrees of freedom of both the edge and entanglement cut are governed by Kramers-Wannier self-dual Hamiltonians, in addition to them being Z2 symmetric, which is imposed by the topological order. Thus, by considering the Wen-plaquette model as a SET, the topological order in the bulk together with the translation invariance of the perturbations along the edge/cut imply an edge-ES correspondence at least in some finite domain in Hamiltonian space.