He may undergo the same accelerations, but at a different place in space, and that makes a difference. The space borne twin observes his sibling (nice touch of gender neutrality, don't you think ) as being very high up in the gravitational well. This mean he perceives the earth clock as running fast relative to his. This is a GR effect, not SR. To quote Stephen Hawking in "A Brief History of Time" (page 33)
Note that Hawking is saying that you get the twin paradox simply by having one twin walk to the top of a mountain. Nothing close to the speed of light or space here. The space and high speed simply make it a larger effect (years instead of nanoseconds). Even these small changes have been observed by taking atomic clocks up in airplanes and observing that the one that flew ran fast compared to the one on the ground.
The theory of relativity gets rid of absolute time. Consider a pair of twins. Suppose that one twin goes to live on the top of a mountain while the other stays at sea level. The first twin would age faster than the second. Thus, if they met again, one would be older than the other. In this case the difference in ages would be very small, but it would be much larger if one of the twins went for a long trip in a spaceship at nearly the speed of light. When he returned, he would be much younger than the one who stayed on Earth. This is known as the twins paradox, but it is a paradox only if one has the idea of absolute time at the back of one's mind. In the theory of relativity there is no unique absolute time, but instead each individual has his won personal measure of time that depends on where
he is and how he is moving.
Before 1915 (note added: This is not a typo, 1915 is the date of the papers on GR), space and time were thought of as a fixed arena in which events took place, but which was not affected by what happened in it. This was true even of the special theory of relativity... The situation, however, is quite different in the general theory of relativity. Space and time are now dynamic quantities: when a body moves, or a force acts, it affects the curvature of space and time---and in turn the structure of space-time affects the way in which bodies move and forces act.
(emphasis is mine)
I will admit that you can compute the outcome of the space thought experiment using only SR, but it requires that you treat the acceleration and the direction change as a sudden shift between two inertial reference frames. That is, you totally ignore what happens during that period. While this use of SR gives you an answer, it does not give you an explanation. That requires general relativity. If you don't believe me, check out the text books. You'll find that they support this position.
"I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind." - William Thompson, 1st Baron Lord Kelvin
"If it was so, it might be, and if it were so, it would be, but as it isn't, it ain't. That's logic!" - Tweedledee
This isn't right. This isn't even wrong. - Wolfgang Pauli