In this paper, we study the problem of estimating the orientation of an event omnidirectional camera mounted on a robot and observing 3D parallel lines in a man-made environment (Manhattan world). We present Gyrevento, the first event-based omnidirectional visual gyroscope. Gyrevento does not require any initialization, provides certifiably globally optimal solutions, and is scalable, since the size of the nonlinear least-squares cost function is independent of the number of lines. Thanks to the Cayley-Gibbs-Rodrigues parameterization of a 3D rotation, this cost function is a degree-four rational function in three variables, which can be efficiently minimized via off-the-shelf polynomial optimization software. Numerical simulations and real-world experiments with a robot manipulator show the effectiveness of our visual gyroscope and elucidate the impact of camera velocity on the attitude estimation error.