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As the gyroscope provides the primary method of input for Cardboard, there isn't much. This post has a few things to do with a phone without gyro and a Cardboard. The only game mentioned is 4 Rooms, which allows you to turn with a controller. You can also look into Trinus VR as mentioned in that thread.
While I personally like the option to use a gamepad for navigation and even rotation, very few people using VR on mobile phones have a gamepad, and even fewer can stomach it, so by requiring it you can severely limit your potential audience.