The P and D surfaces of Hermann Amandus Schwarz have plenty of straight lines on them, and both feature in particular pairs of triangles in parallel planes, twisted to form the top and bottom faces of an antiprism over a regular triangle. The P and D surface then just differ in the height of that antiprism. Another way to think about this is to rotate the cubical fundamental box of the P-surface so that a diagonal becomes vertical.
It is then not surprising that there is an entire 1-parameter family of such surfaces, allowing to deform the P into the D surfaces through embedded minimal surfaces, in contrast to the better known associate family deformation.
Above is the parameter value for the P surface, showing the almost circular tunnels.
When the two triangles are close to each other, there are two different family members with these triangles. Both converge to noded planes. The left image shows catenoidal nodes, placed at the center of the triangles (which is an unstable Plateau solution). The right images shows helicoidal nodes, with two helicoids on each edge of the triangle.
The Weierstrass data of this family are very simple. The quotient of the surfaces under the order 3 rotation are rectangular tori. The picture above shows the divisor of G³(z), which is essentially the Weierstraß ℘-function.