Einstein's special relativity predicts that either twin ages more slowly than the other, as judged from the other twin's system:
David Morin, Introduction to Classical Mechanics With Problems and Solutions, Chapter 11, p. 14: "Twin A stays on the earth, while twin B flies quickly to a distant star and back. [...] For the entire outward and return parts of the trip, B does observe A's clock running slow, but enough strangeness occurs during the turning-around period to make A end up older. Note, however, that a discussion of acceleration is not required to quantitatively understand the paradox..."
Since either twin sees his brother aging more slowly, an absurd conclusion awaits them at the end of the trip unless "enough strangeness occurs during the turning-around period to make A end up older". However the turning-around occurs very far away from the stationary twin so the idea that the turning-around acceleration produces "enough strangeness" and this "enough strangeness" miraculously speeds up the aging of the distant stationary twin is even more idiotic than the original special relativistic conclusion. Still this is not the end of the story - there is a last idiocy that lands the final blow to a student's mind (this mind will never be sane again): Although the turning-around acceleration is crucial (it produces the "enough strangeness"), a discussion of it "is not required to quantitatively understand the paradox".
The turning-around-acceleration idiocy was devised by Einstein in 1918, in a (successful) attempt to camouflage the absurd conclusions of his special relativity (Einstein refers to the acceleration as "gravitational field"):
Albert Einstein 1918: "A homogeneous gravitational field appears, that is directed towards the positive x-axis. Clock U1 is accelerated in the direction of the positive x-axis until it has reached the velocity v, then the gravitational field disappears again. An external force, acting upon U2 in the negative direction of the x-axis prevents U2 from being set in motion by the gravitational field. [...] According to the general theory of relativity, a clock will go faster the higher the gravitational potential of the location where it is located, and during partial process 3 U2 happens to be located at a higher gravitational potential than U1. The calculation shows that this speeding ahead constitutes exactly twice as much as the lagging behind during the partial processes 2 and 4."