Langevin's twin experiment does not lead to a contradiction within the frame of special relativity. Though, such is not the case of one of is variants, the forwards and backwards movement, relative to a Galilean space, of a thin elastic spinning cylinder. The crucial difference is that the space between the 'motionless' part and the one making the journey is always bridged by some continuous portion of the cylinder. Within the frame of special relativity, the study of the cylinder's behaviour and of its changes of shape shows that the experiment can be run with zero intrinsic twist and with intrinsic angular speeds of sections remaining constantly equal to the same value ω. Then, if we calculate the numbers m and n of turns around the axis that two particular points M and N make, we prove n-m is both a zero and a non-zero integer. A world-renowned expert in general relativity wrote the following about this demonstration " I cannot point the author to where the mistake lies"...