What are theoretical computer science jobs? Beside academia which is clearly the home of theorists, I am wondering about industrial jobs related to theoretical computer science, the ones which demand pure mathematical background.

Cheers !
I have a CS Theory background and work in industry doing research at a consulting firm. We get hired by people who want to use various types of computer models and don't have the expertise to create them themselves. This is (partially) a research job because our clients typically have interests usually unaddressed in the academic literature, and so our contracts usually result in us creating new models and techniques that work in their context. I'm currently working on a poster presentation for an academic conference, though one in political science methodology rather than CS.

My group specializes in machine learning and social network analysis, also known as applied linear algebra and graph theory. Our clients tend to be interested in applying this to topics like international relations, logistics, international trade, and cyber security. My coworkers have backgrounds such as a PhD in applied math, a PhD in computational social science, and a masters in computational linguistics. I'm the theoretician in my group and was specifically hired for my pure mathematics knowledge. I primarily work on model design and validation, especially on leveraging graph theoretic structures in the phenomena being modeled to inform model design. I am also the point person for investigating new research and evaluating how it fits into what we are currently working.

I would strongly second Peter Shor's list of recommended skills. The key to being a successful theoretician in industry is to be able to be the bridge between engineers, data analysts, and developers on the one hand and academics on the other.