There is not any official way to download the cube map of your mobile panoramas.
The best lead I can give you is French Photosynth community member, Henri Astre,'s PhotosynthToolkit http://bit.ly/photosynth-toolkit
Currently it only downloads photos in a photosynth, but he has said that a future version may download the six sides of a cube for panoramas as well (the Photosynth mobile app that you've been using makes panoramas, not photosynths).
I know that this doesn't help you for your existing mobile panoramas, but if you shoot photos with your phone's normal camera app or a regular camera, and use Microsoft ICE on Windows http://bit.ly/microsoftice to stitch a panorama to upload here, you can choose to export the panorama as a layered Photoshop file or a flattened JPG, TIFF, BMP, etc. in order to edit it before uploading it to the site.
There is also a panorama publishing plugin for Photoshop. http://bit.ly/pstops