All minimal surfaces can be locally bent in a 1-parameter family of associate minimal surfaces. In the right context this is just a rotation. The best known example is the deformation of the catenoid into the helicoid.
Remarkably, the triply periodic Meeks surfaces, rotated in this sense by 90 degrees, are again in the Meeks family. Very curiously, there are two more known cases where an associate surface of a triply periodic minimal surface is again triply periodic and embedded. One of them is Alan Schoen’s Gyroid, the other is Sven Lidin’s Lidinoid. While the Gyroid occurs in the associate family of the P/D surface, the Lidinoid arises in the associate family of one particular H-surface. To see how this happens, let’s start with a top view of that H surface. When we start rotating in the associate family, the vertices of the three triangles that meet at the center of the image move apart:
But the other vertices then move towards each other, so that, after about 64.2°, they come together:
Not only the vertices fit, but again the surface can be extended by translations in space. Here is a view of a much larger piece.
If you are good at cross-eyed viewing, here is a stereo pair of a side view:
The Inner Frame 