We present an analysis of several high-resolution Chandra grating observations of the X-ray binary pulsar Her X-1. With a total exposure of 170 ks, the observations are separated by years and cover three combinations of orbital and superorbital phases. Our goal is to determine distinct properties of the photoionized emission and its dependence on phase-dependent variations of the continuum. We find that the continua can be described by a partial covering model which above 2 keV is consistent with recent results from Rossi X-Ray Timing Explorer studies and at low energies is consistent with recent XMM-Newton and BeppoSAX studies. Besides a power law with fixed index, an additional thermal blackbody of 114 eV is required to fit wavelengths above 12 Å (~1 keV). We find that likely all the variability is caused by highly variable absorption columns in the range (1-3) × 10²³ cm^–2. Strong Fe K line fluorescence in almost all observations reveals that dense, cool material is present not only in the outer regions of the disk but interspersed throughout the disk. Most spectra show strong line emission stemming from a photoionized accretion disk corona (ADC). We model the line emission with generic thermal plasma models as well as with the photoionization code XSTAR and investigate changes of the ionization balance with orbital and superorbital phases. Most accretion disk coronal properties such as disk radii, temperatures, and plasma densities are consistent with previous findings for the low state. We find that these properties change negligibly with respect to orbital and superorbital phases. A couple of the higher energy lines exhibit emissivities that are significantly in excess of expectations from a static ADC.