  While 2% of active galactic nuclei (AGNs) exhibit narrow emission lines with
line-of-sight velocities that are significantly offset from the velocity of the
host galaxy's stars, the nature of these velocity offsets is unknown. We
investigate this question with Chandra/ACIS and Hubble Space Telescope/Wide
Field Camera 3 observations of seven velocity-offset AGNs at z<0.12, and all
seven galaxies have a central AGN but a peak in emission that is spatially
offset by < kpc from the host galaxy's stellar centroid. These spatial offsets
are responsible for the observed velocity offsets and are due to shocks, either
from AGN outflows (in four galaxies) or gas inflowing along a bar (in three
galaxies). We compare our results to a velocity-offset AGN whose velocity
offset originates from a spatially offset AGN in a galaxy merger. The optical
line flux ratios of the offset AGN are consistent with pure photoionization,
while the optical line flux ratios of our sample are consistent with
contributions from photoionization and shocks. We conclude that these optical
line flux ratios could be efficient for separating velocity-offset AGNs into
subgroups of offset AGNs -- which are important for studies of AGN fueling in
galaxy mergers -- and central AGNs with shocks -- where the outflows are biased
towards the most energetic outflows that are the strongest drivers of feedback.
