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I think you want to use checkboxes, and not radio buttons. The key to this choice is in your requirement that "users need to be able to click any number of choices". Radio buttons, by definition, only allow one of each set to be clicked.
You might also use 80 PAIRS of radio buttons, each pair offering a "yes or no" choice on that category. However, you could also get the same result by having only checkboxes and allowing the visitor to deselect or select categories. Given that you have 80 categories, this would probably be easier to make visually clear.
In either case, you would probably lay this out inside a table with four columns of table cells. Each cell holding one checkbox and its category name, or one pair of radio buttons with their category name.