I don't know of a classical way to achieve that, but I think you can work it out like this:
choose a starting face and assume it to be in the XY plane, with one vertex at the origin and an edge originating from it along the X axis. This is enough to fix the absolute position of the polyhedron.
find two other faces that share this vertex and share the two edges originating from it. Their planes of support form a trihedron, and knowing the three angles at the common vertex, you can find the direction of the third common edge (this will take a bit of spherical trigonometry or vector geometry, presumably).
this allows you to determine the transformation matrix from one face (with a distinguished vertex and distinguished edge) to another.
you can repeat this operation with other triples of faces, each time determining the transformation matrices.
by combining the transformations, you will eventually turn all local coordinate values to global ones.
In a nutshell, you fix one of the faces, then by pulling the right strings you adjust two adjoining faces, then one more, then one more and so on. Every time you will need to find corresponding vertices/edges from other faces and to solve a "trihedron problem", to find a local transformation matrix.
Unless there are many faces or the coordinates are inaccurate, this will reconstruct the whole geometry. In case of inaccuracies, you may have to improve the global model, for instance by least-squares fitting of the edge lengths, using the initial reconstruction. But this is another story.