Hydrothermal synthesis is described of layered lithium iron selenide hydroxides Li1xFex(OH)Fe1ySe (x similar to 0.2; 0.02 < y < 0.15) with a wide range of iron site vacancy concentrations in the iron selenide layers. This iron vacancy concentration is revealed as the only significant compositional variable and as the key parameter controlling the crystal structure and the electronic properties. Single crystal X-ray diffraction, neutron powder diffraction, and X-ray absorption spectroscopy measurements are used to demonstrate that superconductivity at temperatures as high as 40 K is observed in the hydrothermally synthesized samples when the iron vacancy concentration is low (y < 0.05) and when the iron oxidation state is reduced slightly below +2, while samples with a higher vacancy concentration and a correspondingly higher iron oxidation state are not superconducting. The importance of combining a low iron oxidation state with a low vacancy concentration in the iron selenide layers is emphasized by the demonstration that reductive postsynthetic lithiation of the samples turns on superconductivity with critical temperatures exceeding 40 K by displacing iron atoms from the Li1xFex not subset of(OH) reservoir layer to fill vacancies in the selenide layer.