Where can one find jobs where companies hire mathematicians where they can do (not necessarily mainstream) research?
Industrial jobs are announced on different basis than academic ones. Typically, there is no a position to be filled. Rather (larger companies) are hiring on semi-continuous basis, and assign people to groups internally.

Sometimes there are more research-oriented openings, for example Facebook: Data Scientist, Product Science. Just bear in mind it is mostly very applied science, and in fields like Machine Learning, Statistics, maybe some Graph Theory or Cryptography.

So look at big companies (smaller companies may not have such specialized positions or money to invest in long-term development) in technology (Google, Microsoft, Facebook, ...) and their openings. When it comes to direct positions, usually you can choose only after interviews (you rarely see research positions being on the front page).

If it is nice research, or building webpages in a disguise, the only way is to ask people working at the company. Or look at CVs of your colleagues in your field who did internships in these companies (especially ones that you know are pure mathematicians or computer scientists).