I always assumed that you could explain the twin paradox in special relativity using Minkowski diagrams, such as the one shown in wikipedia. In that diagram it is shown how a change in reference frames from moving to the right to moving to the left creates a discontinuity, such that the accelerating observer perceives a sudden jump in time in the clock of the inertial observer. I always assumed that if the change in reference frames was smooth, instead of sudden, the accelerating observer would actually perceive that the clock in the inertial frame runs faster.
But the bottom of the diagram at the origin seems to show a different picture. The simultaneity lines agree for $t<0$, they are horizontal, and for the accelerating observer they change to downwards (from right to left) for $t>0$. It could be argued that if the acceleration is not sudden then this transition from horizontal to downward would be gradual. But if at the start the two coordinate systems are shifted in space (the accelerating one to the right), then it would look like the accelerating observer should see the inertial clock running backwards in time. What am I imagining wrong?
