On studying the Orthogonal Cube, I saw an opportunity to insert six additional struts which would transform the figure into a merger of three six-fold prisms. Although the new struts aren't symmetrical with the existing struts, they combine with the existing ones to form the edges of three six-fold prisms which are shown in red, green and blue in the diagram above. The data for the structure are summarized in the following table:
This design was completed on November 3, 2004. I suppose it could be in Allen Back's and Bob Connelly's Catalogue of Symmetric Tensegrities somewhere, but I haven't been able to find it. This model is also available on the Tensegrity Viewer as "Three 6-Fold Prisms Merged".