Event cameras rely on motion to obtain information about scene appearance. In other words, for event cameras, motion and appearance are seen both or neither, which are encoded in the output event stream. Previous works consider recovering these two visual quantities as separate tasks, which does not fit with the nature of event cameras and neglects the inherent relations between both tasks. In this paper, we propose an unsupervised learning framework that jointly estimates optical flow (motion) and image intensity (appearance), with a single network. Starting from the event generation model, we newly derive the event-based photometric error as a function of optical flow and image intensity, which is further combined with the contrast maximization framework, yielding a comprehensive loss function that provides proper constraints for both flow and intensity estimation. Exhaustive experiments show that our model achieves state-of-the-art performance for both optical flow (achieves 20% and 25% improvement in EPE and AE respectively in the unsupervised learning category) and intensity estimation (produces competitive results with other baselines, particularly in high dynamic range scenarios). Last but not least, our model achieves shorter inference time than all the other optical flow models and many of the image reconstruction models, while they output only one quantity. Project page: https://github.com/tub-rip/e2fai

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