I can't play this game at all on linux, none of the keyboard controls do anything whatsoever.

Same here. As soon as you get back to the level select screen, the controller stops working.

Late to the game but I also have this issue. Running Steam with Arch, and a PS3 controller. I'm using xboxdrv to map the buttons to the xbox controller.

The game works fine until the first level ends, then the controller stops working altogether. When I exit back to Steam, the controller works fine, so I suspect it's an issue in the game.

I've got the exact same problem on Debian stretch.
So I can play this game with an xbox 360 usb controller just fine until I beat the first level. Then the controller seems not to respond anymore. If I launch another mission using the keyboard I realize the controler is like you keep pressing the left arrow.
When I go back to steam in big picture mode. same problem. Controler is like stuck on the left direction and steam goes silly stuck on the left too.
I am using a xbox 360 usb controller.
So I tried both with the defauts xpad driver and then with the xboxdrv.
Both more or less the same annoying bug.
Got the same bug using a logitech controller.

I would really wish to see that fixed in the near future :)