My goal up through grad school had been to spend my life doing academic, research mathematics. I got screwed by my advisor and department, though, so that hasn't worked out. I would nevertheless still like to do math and somehow contribute. Is there any place outside academia that does serious, professional pure math? Ideally, I'd like to at least publish a few research papers on vaguely-mathematical topics, but it's hard to find jobs that even afford that; in fact, it's hard to find jobs that use math beyond the undergrad level. it seems like my best bet is finding something in security related to theoretical computer science, but that's not a field I have any interest in; besides, I'm not sure how interested the industry would be in someone coming in from a pure math background. So, is there any reasonable equivalent or approximation to research mathematics in industry; and, if not, what's the closest I can actually get to it?
This has been here for a while, so I'm not sure it is still a current issue for you. I'm sorry an academic career hasn't worked out to be as you hoped.
If you are serious about commercial sector employment, you need to look at it from a commercial point of view. Some large commercial companies are getting interested in basic research, but it seems to me there is still a clear commercial aspect to their hiring--at least from a long-term point of view. If they are going to pay you \$100K a year, it will cost them at least \$200K a year to have you around (social security tax, health insurance, admin and facilities costs, and so on). So they are asking themselves what you can do for them that makes it worth $200K a year over the long run to have you around.
Many government agencies are similarly task-oriented but in terms that are not always expressed conveniently in dollars (e.g., political objectives, security, etc.). There is much more unadvertised abstract math research in the government sector that you might imagine. The only way to find out is to make applications that mention the full range of your abilities and interests and see how agency recruiters interpret your resume. For example, I can almost guarantee there is a lot of mathematics with security implications for what you call 'theoretical computer science' that does not pop immediately to mind. But publishing results in journals will probably not be involved.
I think you need a fresh perspective and I can't recommend specifics that are sure to be helpful. So generalities: Go to meetings, talk with recruiters, have lunch with nonacademic colleagues about their interests, participate on this site with a 'name' and 'profile' that make you traceable to the persistent, put your (real) name on Linked In (if you can deal with the email traffic that will generate), and so on. One decent 'thread' out of dozens may be all you need.