We analyze new optical spectra of IRAS 20181 - 2244 that confirm its classification as a I Zw 1 object (or narrow-line Seyfert 1 galaxy), one of 20 such objects detected in the ROSAT/IRAS all-sky survey. ASCA X-ray data on IRAS 20181 - 2244 support this classification since they are fitted by a simple power-law spectrum of Gamma = 2.37 +/- 0.09 that is typical of this class but steeper than that of ordinary Seyfert 1 or Seyfert 2 galaxies. Rapid X-ray variability is also observed in IRAS 20181 - 2244, ruling out a hidden/scattered QSO interpretation for this object. Its 0.5 - 10 keV luminosity is 5.5 x 10(44) ergs s(-1) and its variability signifies an efficiency of at least 4 % for conversion of gravitational energy to X-rays. We discuss the important differences between the I Zw 1 class and the "type 2 QSO," or high-luminosity Seyfert 2 galaxy. While I Zw 1 objects of QSO-like luminosity are ubiquitous in infrared, optical, and soft X-ray selected samples, type 2 QSOs are rare, and there is a dearth of evidence for even a single one in X-rays. It is not clear how this imbalance relates to unified models of AGNs.