We study truncated Airy pulses launched into an anomalous dispersion domain of a fiber with strong positive third order dispersion. The pulse quickly reaches the focal point, and then it undergoes a mirror transformation and continues to propagate with the acceleration in the opposite direction. At the focal point all of the light pulse energy is concentrated in a very narrow temporal slot, exhibiting an intriguing pulse compression technique. When both dispersion terms act on the pulse, the focal point extends to a finite area of spreading of the truncated Airy pulse. The size of the area depends on the relative strength of the TOD term relative to its second-order counterpart. After this area, the pulse reemerges again and continues its evolution mirror-transformed. A full exact analytical description of pulses dynamics is developed and verified with direct numerical simulations.