  If space is compact, then a traveller twin can leave Earth, travel back home
without changing direction and find her sedentary twin older than herself. We
show that the asymmetry between their spacetime trajectories lies in a
topological invariant of their spatial geodesics, namely the homotopy class.
This illustrates how the spacetime symmetry invariance group, although valid
{\it locally}, is broken down {\it globally} as soon as some points of space
are identified. As a consequence, any non--trivial space topology defines
preferred inertial frames along which the proper time is longer than along any
other one.
