Since the scene data is embedded in your After Effects .aep file, you usually don't have to save/load Invigorator scenes.
However, for those wishing to load a saved Invigorator scene, you may use Cmd+O (Ctrl+O on Windows) when the 3D Invigorator Startup dialog is shown (the Startup dialog has the "Create 3D Text," "Open Illustrator File," etc. buttons).
You're correct in that you can save a scene while in the Invigorator Set-Up window, but you cannot currently open a scene there. This is a bug that will be fixed in the next version of the Invigorator.