We present an analysis of ionized X-ray disk winds observed in the Fe K band of four stellar-mass black holes observed with Chandra, including 4U 1630-47, GRO J1655-40, H 1743-322, and GRS 1915+105. High-resolution photoionization grids were generated in order to model the data. Third-order gratings spectra were used to resolve complex absorption profiles into atomic effects and multiple velocity components. The Fe XXV line is found to be shaped by contributions from the intercombination line (in absorption), and the Fe XXVI line is detected as a spin-orbit doublet. The data require 2-3 absorption zones, depending on the source. The fastest components have velocities approaching or exceeding 0.01c, increasing mass outflow rates and wind kinetic power by orders of magnitude over prior single-zone models. The first-order spectra require re-emission from the wind, broadened by a degree that is loosely consistent with Keplerian orbital velocities at the photoionization radius. This suggests that disk winds are rotating with the orbital velocity of the underlying disk, and provides a new means of estimating launching radii -- crucial to understanding wind driving mechanisms. Some aspects of the wind velocities and radii correspond well to the broad-line region (BLR) in active galactic nuclei, suggesting a physical connection. We discuss these results in terms of prevalent models for disk wind production and disk accretion itself, and implications for massive black holes in active galactic nuclei.