The back is a single piece of ½ MDO plywood, 36 by 76 as seen on the exterior plans. The surface decoration is ½ MDF fibreboard. This stuff produces nasty carcinogenic dust when cut with a table saw, so use a mask, extractor and do it outside if you can.
The cross pieces are bevelled inward on the long edge and outward on the short edge so that they meet the full height boards at an angle. I cut them slightly longer than necessary then trimmed them to the exact width, as even a 1/16 gap is very obvious. I still ended up using spackle to fill a few gaps. I glued the MDF boards directly onto the MDO. Since they arent structural Im not expecting this to cause a problem, provided the glue holds (!).
I glued a 1 ½ by 3/8 piece of wood down the centre of the back to resemble the doorjamb seen on the front.