Trapezoid Order Classification
  We show the nonequivalence of combinations of several natural geometric restrictions on trapezoid representations of trapezoid orders. Each of the properties unit parallelogram, unit trapezoid and proper parallelogram, unit trapezoid and parallelogram, unit trapezoid, proper parallelogram, proper trapezoid and parallelogram, proper trapezoid, parallelogram, and trapezoid is shown to be distinct from each of the others. Additionally, interval orders are shown to be both unit trapezoid and proper parallelogram orders. 
Stephen Ryan