I have seen purple tabards at Gulf Wars with the gold cross on the
front and the gold falcon on the back.
I think you may be mis-remembering slightly. There are tabards with the Calon Cross on the front and back, and those are for the Royal Guard. There are also tabards with the striking falcon on the front and back, and those are what the army (all parts thereof) wear.
I do not recall ever seeing a tabard with the falcon on one side and the cross on the other. That doesn't mean they don't exist, but they haven't registered in my consciousness.