The disorganized fluctuations of turbulence are crucial in the transport of particles or chemicals1, 2 and could play a decisive role in the formation of rain in clouds3, the accretion process in protoplanetary disks4, and how animals find their mates or prey5, 6. These and other examples7 suggest a yet-to-be-determined unifying structure of turbulent flows8, 9. Here, we unveil an important ingredient of turbulence by taking the perspective of an observer who perceives its world with respect to three distant neighbours all swept by the flow. The time evolution of the observer’s world can be decomposed into rotation and stretching. We show that, in this Lagrangian frame, the axis of rotation aligns with the initially strongest stretching direction, and that the dynamics can be understood by the conservation of angular momentum. This ‘pirouette effect’ thus appears as an important structural component of turbulence, and elucidates the mechanism for small-scale generation in turbulence.