You've always been able to use any controller you want when playing Google Stadia with a PC, phone, or tablet, but not with a Chromecast or other similar setups. Google is finally changing this with Tandem Mode, a new Stadia feature that turns the Type-C connector on the wireless Stadia Controller into an input.
"When you link your Stadia Controller wirelessly to Stadia," Google wrote in a new support article, "you can plug a secondary controller directly into its USB-C port. When you press buttons on the secondary controller, it will act as though the primary controller's buttons are being pressed. [...] You can use either controller to play games on Stadia, or both at the same time."
This new functionality has two main advantages. First, it allows you to use select third-party controllers with Stadia, regardless if they can be connected to the device you're playing Stadia on. Second, it's easier for people with physical disabilities to play Stadia, since Google is supporting the Xbox Adaptive Controller. If the linked controller has a Type-C connector, you can just use a regular C-to-C cable, otherwise you'll need an adapter or special cable.
Methods of connecting the Stadia Controller to third-party controllers
There's one catch: Google is whitelisting the controllers you can use, so don't expect the same number of options as you could use with a traditional Android device or PC. The support page lists the Stadia Controller, Xbox Adaptive Controller, Xbox One Controller, Xbox One Elite Controller (Series 1 and 2), and Sony DualShock 4 Controller as the only options right now.
It would be great to see Google add support for some inexpensive third-party controllers (or controllers from previous-gen consoles), and racing wheels could be great for titles like F1 2020. There's also no option to use a connected gamepad for a second player, at least not yet.
- Google Support