We present and analyze ROSAT PSPC observations of eight Seyfert 2 galaxies, two Seyfert 1/QSOs, and one IR-luminous non-Seyfert galaxy. These targets were selected from the Extended 12 μm Galaxy Sample and, therefore, they have different multiwavelength properties from most (optically or X-ray-selected) Seyfert galaxies previously observed in the soft X-rays. The targets were also selected as having atypical X-ray fluxes among their respective classes, e.g., relatively X-ray-strong Seyfert 2s and X-ray-weak Seyfert 1/QSOs. Comparing our observations with those from the ROSAT All-Sky Survey, we find variability (of a factor of 1.5-2 in flux) in both the Seyfert 1/QSOs, but in none of the Seyfert 2s. Both variable objects have steeper photon indices in the more luminous state, with the softest (< 1.0 keV) flux varying the most. The timescales indicate that the variable component arises from a region less than a parsec in size. Fitting the spectra to an absorbed power-law model, we find that both the Seyfert 2s and the Seyfert 1/QSOs are best fit with a photon index of 3.1-3.2. This is in agreement with the average photon index of a sample of Markarian Seyfert 2s observed by Turner, Urry, & Mushotzky, indicating that most Seyfert 2s, even those displaying a wide variety multiwavelength of characteristics, as well as some Seyfert 1/QSOs, have a photon index much steeper than the canonical (Seyfert 1) value of ∼1.7. One possible explanation is that these objects have a flatter continuum plus a soft (< 1.0 keV) excess in the form of high-EW iron and/or oxygen fluorescence lines, a blackbody or even a thermal plasma. Alternatively, the underlying continuum may indeed be steep, powered by a different physical mechanism than that which produces the flat continua in other Seyfert 1/QSOs. We imaged one Seyfert 2 (NGC 5005) with the ROSAT HRI, finding about 13% of the soft X-rays to come from an extended source. This object also has the most evidence from spectral fitting for an extra contribution to the soft X-ray flux in addition to a power-law component, indicating that different components to the soft X-ray spectrum of this object (and likely of other X-ray-weak Seyfert galaxies) may come from spatially distinct regions. © 1996. The American Astronomical Society. All rights reserved.