This is a form of the Rhombicuboctahedron
To make this polyhedron you will need the four templates pictured below.
To see the full-sized template click on this image:
The squares will be made with 3 colors. Just like the cube, opposite squares will be made of the same color. Below those square faces and in a parallel plane, you will create an octahedron of the same color. (In the picture above you can see a yellow octahedron facing slightly to the right with a yellow square face above it but in a parallel plane.)
The equilateral triangle faces are made of two different but alternating colors.
Each of the right triangular prism indentations maintain sides and end pieces that carry the color of each parallel plane.
Begin your construction with one square piece (cube face) alternately surrounded by rectangles of the other two cube colors. Build the side, irregular tetrahedral piece from one equilateral triangle face surrounded by 3 right triangle faces. Figure out the appropriate colors by using the colors of what will become parallel faces. Attach that tetrahedral piece and construct the other 3 corner, tetrahedral pieces in the same manner.
You should now be able to assemble the polyhedron by visualizing parallel planes and opposite cubic faces.