We employ measurements of the [alpha/Fe] ratio to separate them into likely thin- and thick-disk subsamples. Both subsamples exhibit strong gradients of orbital rotational velocity with metallicity, but with opposite signs (-20 to -30 km/s/dex for the thin-disk population, and +40 to +50 km/s/dex for the thick-disk population). We find that the rotational velocity decreases with the distance from the plane for both disk components. The mean rotational velocity is uncorrelated with Galactocentric distance for the thin-disk subsample, and exhibits only a marginally significant correlation for the thick-disk subsample. Thick-disk stars exhibit a very strong trend of orbital eccentricity with metallicity (-0.2 dex^-1), while the eccentricity does not change with metallicity for the thin-disk subsample. The eccentricity is almost independent of Galactocentric radius for the thin-disk stars, while a marginal gradient of the eccentricity with distance exists for the thick-disk population. Both subsamples possess similar trends of increasing eccentricity with distance from the Galactic plane. The shapes of the overall distributions of orbital eccentricity for the thin- and thick-disk populations are quite different from one another, independent of distance from the plane; neither subsample has significant numbers of stars with eccentricity above 0.6. These observational results provide strong new constraints on models for the formation and evolution of the Milky Way's disk system.