We consider three-dimensional cubic barcodes, consisting of smaller cubes, each built from one of two possible materials and carry one bit of information. To retrieve the information stored in the barcode, we measure a 2-D projection of the barcode using a penetrating wave such as X-rays, either using parallel-beam or cone-beam scanners from an unknown direction. We derive a theoretical representation of this scanning process and show that for a known barcode pose with respect to the scanner, the projection operator is linear and can be easily inverted. Moreover, we provide a method to estimate the unknown pose of the barcode from a single 2-D scan. We also propose coding schemes to correct errors and ambiguities in the reconstruction process. Finally, we test our designed barcode and reconstruction algorithms with several simulations, as well as a real-world barcode acquired with an X-ray cone-beam scanner, as a proof of concept.