The way I see it (at least for our purposes) alkaline is simply a bad confusing word. Instead we should use the words acid, basic and buffering capacity. (Chemists feel free to set me straight because when it comes to alkalinity is am so lost!)

R/O water when spanking new is close to PH 7. Soon it absorbs carbon dioxide from the air and forms carbonic acid. The PH then shifts to about 6. However, as Demoreal pointed out, its buffering capacity is very weak. Any addition of a strong acid or base will easily shift the PH up or down.

In order to test pure water use a freshwater aquarium test kit (the liquid drops). They are $5 at Walmart for 250-750 tests.