Neuromorphic imaging reacts to per-pixel brightness changes of a dynamic scene with high temporal precision and responds with asynchronous streaming events as a result. It also often supports a simultaneous output of an intensity image. Nevertheless, the raw events typically involve a great amount of noise due to the high sensitivity of the sensor, while capturing fast-moving objects at low frame rates results in blurry images. These deficiencies significantly degrade human observation and machine processing. Fortunately, the two information sources are inherently complementary -- events with microsecond temporal resolution, which are triggered by the edges of objects that are recorded in latent sharp images, can supply rich motion details missing from the blurry images. In this work, we bring the two types of data together and propose a simple yet effective unifying algorithm to jointly reconstruct blur-free images and noise-robust events, where an event-regularized prior offers auxiliary motion features for blind deblurring, and image gradients serve as a reference to regulate neuromorphic noise removal. Extensive evaluations on real and synthetic samples present our superiority over other competing methods in restoration quality and greater robustness to some challenging realistic scenarios. Our solution gives impetus to the improvement of both sensing data and paves the way for highly accurate neuromorphic reasoning and analysis.