Quote by edpell
The folks left behind are moving at exactly the same speed as the folks in the ship (when seen from the ship) (I say speed not velocity, the velocities are equal and opposite).

This is simply not true. The coordinate velocity of earth in the ship's frame doesn't match the velocity of the ship in earth's frame for the entire trip.
In the ship's frame(s), earth's coordinate distance from the ship increases at a constant rate, then during the turnaround goes from the length contracted distance to the proper distance and back again in a very short time, then decreases at a constant rate back to zero.
This is very different from the ship's coordinate distance from earth in earth's frame, which just increases at a constant rate then decreases at a constant rate.
Maybe someone could graph the coordinate distance between the earth and ship against each twin's clock reading to show how different they are.