It is well known that electron emission can restructure the thin sheaths at plasma-facing surfaces. But conventional models assume that the plasma's structure negligibly changes (the “presheath” is still thought to be governed by ion acceleration to the Bohm speed). Here, it is shown by theory and simulation that the presheath can take a fundamentally different structure where the emitted electrons entering the quasineutral region cause numerous changes. Gradients of total plasma density, ion and electron pressures, and electric potential throughout the “inverted” presheath can carry different magnitudes, and opposite signs, from Bohm presheaths.