Continued exploration revealed a very simple way of orienting eight chestahedra together:
(Click any image for a larger version.)
This form is made of two identically oriented sets of four chestahedra inverted with respect to each other. Take one set of four and invert it front to back, as in a mirror, and you get the other set of four. They fit together perfectly to create a double, inverted cross.
The side view shows how the chestahedra are fit together. The octagonal perimeter of the chestahedra is not twisted, but lies in a plane. The inside space of this form consists of eight kite-shaped faces alternating directions, in a tube-like or ovoid shape.
I am not aware of any types of crosses that are formed through the use of a three-dimensional form, let alone in a way that the form itself comes about as a mirror image of itself. The two cross shapes are a necessary result of the simple way in which the chestahedra are placed together; what a fascinating form to yield so many interesting shapes!
The cross, of course, has an extensive history, and there are a huge variety of forms, but a little research turned up some interesting tidbits…