OVERVIEW: At time zero, one twin stays on Earth and the other twin starts his travel into space at a constant speed of 0.8 x c. After 25.0 Earth years, the traveling twin instantly reverses direction (switches to a returning space ship). After another 25.0 Earth years, the traveling twin reaches Earth and has aged only 30.0 years while his Earth-bound twin has aged 50.0 years.
FROM EARTH FRAME: Since the Earth-bound twin "sees" the moving space ship clocks (outgoing and incoming) each ticking at only 0.6 times the rate of his own Earth clock, then after 50 Earth years, only 30 space ship years will have passed.
FROM SPACESHIP FRAMES: The twin on the outgoing and then the incoming space ship (two separate frames) always sees the moving Earth clock ticking at only 0.6 times the rate of the space ship clock. That's why on each space ship, 15.0 space ship years pass while only 9.0 Earth years pass.
THE "LOST" 32.0 EARTH YEARS: At the journey's mid-point, outgoing space ship time is 15.0 years while only 9.0 Earth years will have passed. Once the twin instantly moves to the incoming space ship, he resets his clock to 15.0 years but, to his surprise, he sees that the Earth clock reading has jumped from 9.0 years to 41.0 years! General relativity explains this jump as follows: The huge acceleration of the transfer causes the spaceship twin to see his own clock slow down and to see the inertial clock on Earth to speed up by 32.0 years. In this sense, the forces of the spaceship transfer, however brief in time, "thrusts" the traveling twin 32.0 years into Earth's future.