In the United States, it is definitely possible to do mathematical research at some industry labs, and publish/collaborate just as an academic would. (This is much easier if your research straddles math and computer science, but still possible in some other cases.) My employer, IBM Research, has labs in the New York and San Francisco areas which hire mathematicians to do research with a strong computer science slant. Also, Microsoft Research in Redmond, WA, Mountain View, CA, and Cambridge, MA (and the other Cambridge!) hires people with similar background. Google and Yahoo also have labs but I am not sure they do as much theoretical work.
In the San Francisco area, there are also government-affiliated labs like Sandia, Lawrence Berkeley, and Lawrence Livermore. However I am not sure how much of their work they can publish, and what kind of security clearance they need to work on interesting projects. (They do publish at least some papers, though.)
Every group has different foci, so your chances of working in industry really depend on how much your interests and work intersect with some group. It will require some effort on your part to find appropriate places for you.
Note the above groups are just those that I am personally familiar with. Also, the above labs hire postdocs as well (in fact I am currently one).