If you have a VRML plugin you can see and investigate the complete polyhedron here. (You can check here and find available plugins for your browser/OS.) The polyhedron is a compound of 12 cubes and belongs to the symmetry group 12|S4xI|D1xIμ, as described in H.F.Verheyen's Symmetry Orbits. The angle μ uniquely identifies one compound in the group and may vary between 0 and 45 degrees (with 0 and 45 not included). For some angles μ the compound has some special properties. For this compound the angle μ=(1/2)atan((4/7)√2) and for that angle the compound can be divided into three Bakos compounds (4|S4xI|D3xI) For each Bakos compound holds that one of the fourfold axes is shared with a fourfold axis of the compound; the other two axes share a twofold axis with the compound.
This compound can be seen as a twin of this compound, since it can be interpreted as a multiplication between Bakos' compound and the classic compounds of 3 cubes, where the twin compound can be seen as a multiplication between the Classic compound and Bakos.
Here are some more pictures of the model:
A complete table about cube compounds can be found here here.
If you are interested, here is a document that describes how the templates were calculated. It also contains the templates and the adjusted templates that I am using, since some faces became so tiny that is was not very practical to build an exact model. Here you can see how I built the model.