You can, but the two ends don't always grind up very well. And stringey things like banana peels and celery will tend to cause it to seize up. I would recommend using them in your garden for compost. They are supposed to be really good for tomato and rose plants.
It has been not recommended, but I myself have cut 'em up and experimented with it, and it seems to work just find. I do know that egg shells and ice, either by themselves or together, sharpen the blades of a garbage disposal quite well, and it helps to do it daily depending on how much you use the garbage disposal.