Theoretical considerations, observations, and simulations have shown that the B-y component of the interplanetary magnetic field (IMF) may cause twisting of the magnetotail. However, the fundamental issues, the temporal and spatial responses of the magnetotail in the twisting process, are still unresolved. We report unique multipoint observations of the response of the magnetotail to the variations in IMF B-y on 1-2 January 2009. For the first time, estimates of the tail twisting response time at different (Time History of Events and Macroscale Interactions during Substorms, THEMIS) distances in the same event are inferred. Using cross-correlation and timing analyses, we find that the tail twisting propagates from farther out toward the Earth and the response time increases significantly to the inner magnetosphere.