  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 super-orbital 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 \rxte
studies and at low energies is consistent with recent \xmm and \sax studies.
Besides a powerlaw with fixed index, an additional thermal blackbody of 114 eV
is required to fit wavelengths above 12 \AA ($\sim$ 1 keV). We find that likely
all the variability is caused by highly variable absorption columns in the
range (1 -- 3)$\times 10^{23}$ 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. 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 super-orbital phases.
A couple of the higher energy lines exhibit emissivities that are significantly
in excess of expectations from a static accretion disk corona.
