Molecular oxygen-enriched water (OxEW) is advocated in popular media as useful for various health issues, presumably due to involvement of a purported antioxidant activity and to such notions as "active oxygen." To our knowledge, there are no explicit reports in the scientific literature where such redox reactivity would be described and explained. Reported here are data showing that a commercial preparation of OxEW does display a measurable, albeit very small, antioxidant activity as monitored by reaction with a standard reagent, DPPH. Moreover, OxEW also displays an apparent pro-oxidant reactivity, against caffeic acid. This does not correlate with any UV-vis-detectable contents of chemical substances in the water, nor can it be explained by typical chemical impurities (e.g., hydrogen peroxide or molecular hydrogen) that would arise upon enrichment with molecular oxygen of pure water by the two most common procedures: purging with gaseous O2 or electrolysis. Instead, this apparent redox reactivity is revealed to be due to differences in pH and in chemical content - and the differences in turn are most likely due to the trace amounts of inorganic ions/elements in the OxEW; importantly, electrolysis, which is often employed as a means to generate O2 in OxEW preparation, is also found to enhance the redox effect of OxEW-like preparations. Thus, in line with expectations, the herein-reported data show that there are no long-lived reactive oxygen species, no activated oxygen, and no extra reducing agents in OxEW - but that an apparent weak redox reactivity can still be measured and assigned to simple side effects of the electrolysis procedure presumably performed in order to enrich the sample in oxygen.