  We analyze observations obtained with the Chandra X-ray Observatory of bright
Compton thick active galactic nuclei (AGNs), those with column densities in
excess of 1.5 x 10^{24} cm^{-2} along the lines of sight. We therefore view the
powerful central engines only indirectly, even at X-ray energies. Using high
spatial resolution and considering only galaxies that do not contain
circumnuclear starbursts, we reveal the variety of emission AGNs alone may
produce. Approximately 1% of the continuum's intrinsic flux is detected in
reflection in each case. The only hard X-ray feature is the prominent Fe K
alpha fluorescence line, with equivalent width greater than 1 keV in all
sources. The Fe line luminosity provides the best X-ray indicator of the unseen
intrinsic AGN luminosity. In detail, the morphologies of the extended soft
X-ray emission and optical line emission are similar, and line emission
dominates the soft X-ray spectra. Thus, we attribute the soft X-ray emission to
material that the central engines photoionize. Because the resulting spectra
are complex and do not reveal the AGNs directly, crude analysis techniques such
as hardness ratios would mis-classify these galaxies as hosts of intrinsically
weak, unabsorbed AGNs and would fail to identify the luminous, absorbed nuclei
that are present. We demonstrate that a three-band X-ray diagnostic can
correctly classify Compton thick AGNs, even when significant soft X-ray line
emission is present. The active nuclei produce most of the galaxies' total
observed emission over a broad spectral range, and much of their light emerges
at far-infrared wavelengths. Stellar contamination of the infrared emission can
be severe, however, making long-wavelength data alone unreliable indicators of
the buried AGN luminosity.
