There exist many problem domains where the interpretability of neural network models is essential for deployment. Here we introduce a recurrent architecture composed of input-switched affine transformations - in other words an RNN without any explicit nonlinearities, but with input-dependent recurrent weights. This simple form allows the RNN to be analyzed via straightforward linear methods: we can exactly characterize the linear contribution of each input to the model predictions; we can use a change-of-basis to disentangle input, output, and computational hidden unit subspaces; we can fully reverse-engineer the architecture's solution to a simple task. Despite this ease of interpretation, the input switched affine network achieves reasonable performance on a text modeling tasks, and allows greater computational efficiency than networks with standard nonlinearities.