OK, I am not an engineer, but I have studied Engineering a bit, and I have worked with engineering issues for a long time, so I think I understand the basic principles here. One of the first problems we have is the terminology here. What we are talking about “cross bracing” is more often termed “wind bracing,” since it is the function of this bracing to resist lateral loads (i.e. the wind). All structures need to resist lateral loads or they will fall down in the first wind storm. Conventional steel high rise buildings usually do this through the use of a stiffened core. Some cores are stiffened through the use of diagonal (or cross bracing) between the columns to form a vertical truss. Some buildings stiffen the core by building it out of concrete and using masonry infill between the columns. (i.e. the Windsor tower).
Some buildings use the exterior walls to provide wind bracing. The John Hancock building in Chicago has the diagonal bracing on the exterior of the building, giving it its distinctive look.
For the WTC towers, the wind bracing was also provided by the exterior walls. The close set columns with those wide spandrel plates between them formed what is known as a Vierendeel truss. Each exterior wall was extremely stiff in the plane of the wall. The floor slabs kept the walls in alignment to each other and redistributed the lateral loads from wall to wall.
On those floors were Vierendeel trusses existed on the exterior walls, there was no cross (or diagonal) bracing in the core structure.
Does this mean that there was not cross (or diagonal) bracing in the core area? Actually, no it doesn’t. There was, in fact cross bracing present in specific areas of the building.
First was the hat truss at the top of the structure. This bracing was intended to tie the exterior and the core elements of the structure together and it served an important role in redistributing the horizontal and lateral loads on the building. This bracing extended from the core to the exterior walls.
There are two other key locations of cross bracing, just within the core structure. Cross bracing was installed in those locations where the Vierendeel truss did not exist on the outside of the building. The first and most important was in the bottom seven floors of the building, below the trident split of the exterior columns. There were no spandrel plates in this area. These trusses are visible in the picture in the OP of this thread (not the crane towers)
The other location of cross bracing in the core area was on the mechanical floors. There were no exterior spandrel plates on these floors to accommodate the air intakes for the HVAC systems.
So to answer the question: was there cross bracing in the core, the answer is yes and no.
Now can we let this thread die?