  In a multiply connected space, the two twins of the special relativity twin
paradox move with constant relative speed and meet a second time without
acceleration. The twins' situations appear to be symmetrical despite the need
for one to be younger due to time dilation. Here, the suggestion that the
apparent symmetry is broken by homotopy classes of the twins' worldlines is
reexamined using space-time diagrams. It is found that each twin finds her own
spatial path to have zero winding index and that of the other twin to have
unity winding index, i.e. the twins' worldlines' relative homotopy classes are
symmetrical. Although the twins' apparent symmetry is in fact broken by the
need for the non-favoured twin to non-simultaneously identify spatial domain
boundaries, the non-favoured twin {\em cannot} detect her disfavoured state if
she only measures the homotopy classes of the two twins' projected worldlines,
contrary to what was previously suggested. We also note that for the
non-favoured twin, the fundamental domain can be chosen by identifying time
boundaries (with a spatial offset) instead of space boundaries (with a temporal
offset).
