One Category, Multiple Templates

I’ve got a page, which has a two links. Each leads to the same category of posts, but I want one of those links to open up and display those posts in one format, and then I want the other link to display that same content/category in another format.

I’m currently using category-6.php to style the first link, but can’t seem to figure out how to specify a different template for the second link.

Any ideas? I couldn’t find anything in the codex.