We present the results of ROSAT PSPC observations of three active galactic nuclei (AGN) with extremely strong Fe ii emission (PHL 1092, IRAS 07598 + 6508 and I Zw 1) and two AGN with very weak Fe ii emission (Mrk 10 and 110). The weak Fe ii emitters have X-ray spectra typical of Type 1 AGN (α = 1.35 and 1.41, where α is the spectral energy index). Of the strong Fe ii emitters, two have steep spectra (PHL 1092 has α = 3.5, and I Zw 1 has =2.0) and the third, IRAS 07598 + 6508, is barely detected and so is extremely X-ray-quiet (αox=2.45). During our observations, PHL 1092 varied by a factor of 4, unusually fast for such a high-luminosity object, and requiring an efficiency of matter-to-energy conversion of 2 per cent or more. Compiling recently published data on other strong Fe ii emitters, we find that they are always X-ray-quiet, and usually X-ray-steep. Adding these data to the complete UVX-selected quasar sample of Laor et al., we find a statistical connection of Fe ii/Hβ with αx but not a simple relationship: weak Fe ii emitters always have flat spectra, but strong Fe ii emitters can be either flat or steep. A much cleaner relationship exists between Fe ii strength and X-ray loudness, as quantified by αix, the spectral index between 1 μm and 2 keV. We also confirm that Fe ii/Hβ anticorrelates with Balmer line velocity width, which in turn correlates well with both αx and αix in the sense that AGN with narrow lines are X-ray-quiet. There is also marginal evidence that Fe ii/Hβ correlates with both optical continuum slope and the curvature of the optical-UV-X-ray continuum: strong Fe ii objects tend to have steeper continua and weaker ‘blue bumps’. The amount of extinction required to explain the optical steepening compared to normal quasars [E(B − V) in the range 0.2 to 0.6] suggests absorbing columns in the range (1 – 3) × 1021 cm−2, just about the right amount to reduce the ROSAT-band X-ray flux by enough to explain the correlation with αix. However, the spectral shapes observed in the ROSAT band are not consistent with a simple absorption model. Three objects in our total sample of 19 stand out persistently in all correlations: Mrk 231, IRAS 07598 + 6508 and Mrk 507. Interestingly, two out of the three are known to have low-ionization, broad absorption lines in the UV, and the third (Mrk 507) has no UV spectrum available. Furthermore, low-ionization, broad absorption lines are at least an order of magnitude more common in strong Fe ii emitters than in quasars in general. Overall, continuum shape and blueshified absorption should be added to the intriguing cluster of properties which all vary loosely together, and which has been isolated as ‘eigenvector 1’ by Boroson & Green: Fe ii strength, velocity width, narrow-line strength and line asymmetry. We suggest that the underlying parameter is the density of an outflowing wind.