Two surveys of dynamical models of highly flattened, triaxial elliptical galaxies with isothermal potentials have been constructed. These models were constructed in order to better understand the range of possible observable dynamical properties of triaxial galaxies. All models have been constructed so that they appear as E6 galaxies when seen from their intermediate axes. However, one set of models is nearly oblate; the other is nearly prolate. the models are constructed with massive halos such that M/L ∝ r. Triaxial models of either shape can be constructed with their projected axes of rotation at any position angle with respect to the major axes of the galaxies. the most surprising result is that in most models, the position angle of maximum observed rotation is not perpendicular to the position angle of zero rotation.