The mechanisms that underlie fascinating inter-individual interactions among animal groups have attracted increasing attention from biologists, physicists, and system scientists. There are two well-known types of interaction patterns: hierarchical and egalitarian. In the former type, individuals follow their leaders, whereas they follow their neighbors in the latter. Using high-resolution spatiotemporal data derived from the free flights of a flock of pigeons, we show that pigeon flocks actually adopt a mode that switches between the two aforementioned strategies. To determine its flight direction, each pigeon tends to follow the average of its neighbors while moving along a smooth trajectory, whereas it switches to follow its leaders when sudden turns or zigzags occur. By contrast, when deciding how fast to fly, each pigeon synthesizes the average velocity of its neighbors. This switching mechanism is promising for possible industrial applications in multi-robot system coordination, unmanned vehicle formation control, and other areas.