Dihedral tilings

Problem: Which sets of exactly \(2\) shapes can be used to tile the plane?

Known results:

The semi-regular tessellations are the only dihedral tilings involving regular polygons.

The Koch snowflake admits a dihedral tiling.

Replace each of the three types of lines, in an equilateral triangle tiling, with paths that repeat, as shown below. Assuming no pair of paths cross, you'll end up with a dihedral tiling.

Every rep-tile forms a dihedral tiling. Just choose the smaller shape, and any larger copy. It's easy to see these will form a dihedral tiling.