(Abridged) We compared the quantified morphological properties of 37 intermediate redshift MgII absorption selected galaxies to the properties of the absorbing halo gas, observed in the spectra of background quasars. The galaxy morphologies were measured using GIM2D modeling of Hubble Space Telescope WFPC-2 images and the absorbing gas properties were obtained from HIRES/Keck and UVES/VLT quasar spectra. We found a 3.1 sigma correlation between galaxy morphological asymmetries normalized by the quasar-galaxy projected separations, A/D, and the MgII rest-frame equivalent widths. Saturation effects cause increased scatter in the relationship with increasing W_r(2796). We defined a subsample for which the fraction of saturated pixels in the absorption profiles is f_sat<0.5. The correlation strengthened to 3.3 sigma. We also find a paucity of small morphological asymmetries for galaxies selected by MgII absorption as compared to those of the general population of field galaxies, as measured in the Medium Deep Survey. The K-S probability that the two samples are drawn from the same galaxy population is ruled out at a 99.8% confidence level. The A/D-W_r(2796) correlation suggests a connection between the processes that perturb galaxies and the quantity of gas in their halos, normalized by the impact parameter. Since the perturbations are minor, it is clear that dramatic processes or events are not required for a galaxy to have an extended halo; the galaxies appear normal. We suggest that common, more mild processes that populate halos with gas, such as satellite galaxy merging, accretion of the local cosmic web, and longer-range galaxy-galaxy interactions, consequently also induce the observed minor perturbations in the galaxies.
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