In an effort to understand better the formation and evolution of barred galaxies, we have examined the properties of equatorial orbits in the effective potential of one specific model of a rapidly rotating, steady state gasdynamical bar that has been constructed via a self-consistent hydrodynamical simulation. At a given value of the Jacobi constant, roughly half of all test particles (stars) that are injected into the equatorial plane of this potential follow quasi-ergodic orbits; most regular prograde orbits have an overall bow tie shape; and some trace out trajectories that resemble the x4 family of regular, retrograde orbits. The bow tie orbits appear to be related to the 4/1 orbit family discussed by Contopoulous in 1988, but particles moving along a bow tie orbit pass very close to the center of the bar twice each orbit. Unlike the barlike configurations that previously have been constructed using dissipationless, N-body simulation techniques, the effective potential of our gasdynamical bar is very shallow and generally does not support the x1 family of orbits. If primordial galaxies evolve to a rapidly rotating barlike configuration before a significant amount of star formation has taken place and then stars form from the gas that makes up the bar, the initial stellar distribution function should consist of orbits that are (1) supported by the gaseous barlike potential and (2) restricted to have initial conditions dictated by the gasdynamics of the bar. With this restriction hypothesis in mind, we propose that stellar dynamical systems that form from gaseous bars will have characteristics that differ significantly from systems that form from a bisymmetric instability in an initially axisymmetric stellar system. Since bow tie orbits are preferred over x1 orbits, for example, such systems should have a more boxy or peanut shape when seen face-on; there will be a mechanism for funneling material more directly into the center of the galaxy; and, near the galaxy center, stars may appear to move along retrograde trajectories.