I see, on many forums and other places, many questions asked by many intelligent people, struggling with the concepts of time dilation, space contraction, "light" speed constancy etc.
No symmetry is indeed a common argument, but it's almost always given as a superficial explanation: of course there is no problem, since the two situations are different.
But thinking further, it's not that simple, especially when trying to explain what is happening, concretely and physically, in the traveller's reality.