We calculate the self-consistent electronic structure of GaAs-${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As split-gate quantum wires taking into account the ordering in the donor layer which occurs when the ionization fraction F is less than unity. Ordered ion distributions are generated via a Monte Carlo simulation and show glassy behavior as well as formation of a Wigner-like crystal without long-range order for sufficiently small F. We study the structure of the wire modes and show that ordering substantially reduces the standard deviation of mode energies, making single-mode wires possible. \textcopyright{} 1996 The American Physical Society.