Just like what Chuang has said in previous comment, if we align the (111) direction of the gyroidal surface in one unit cell vertically and then view it from the side, it is easy to see that the whole structure of this unit cell exactly consists of three layer. The center of each layer is a monkey saddle.
I found something similar discussed in a paper by Klinowski (Exact computation of the triply periodic G minimal surface, CPL, 321, 363-371, 2000) in which the authors mentioned the so-called fundamental patches, the smallest portion of the surface from which the entire surface can be constructed. Similar concept can be found in the description of a crystal. Instead, here we have a patch of surface. Klinowski also pointed out in his paper that a unit cell of the G surface consists of 12 fundamental patches. I don't have Chuang's beaded model with me now. It would be interesting to check it.