it's not fixed, but I always thought retro-direct drives were interesting. Pedal forwards and you have one gear, pedal backwards and it is a different gear.

as for why you can't shift with a fixed it has to do with how long it would take a change in pedal movement (when you resist for instance) to translate back to the real wheel, assuming that you have it absolutely dialed in so that you don't need to worry about your chain falling off.