We classify differentiable structures on a line \(\mathbb {L}\) with two origins being a non-Hausdorff but \(T_1\) one-dimensional manifold obtained by “splitting” \(0 \in \mathbb {R}\) into two points 0 and \(\bar{0}\) . For \(k\in \mathbb {N}\cup \{\infty \}\) let H be the group of homeomorphisms h of \(\mathbb {R}\) such that \(h(0)=0\) and the restriction of h to \(\mathbb {R}\setminus 0\) is a \(\mathcal {C}^{k}\) -diffeomorphism. Let also D be the subgroup of H consisting of \(\mathcal {C}^{k}\) -diffeomorphisms of \(\mathbb {R}\) also fixing 0. It is shown that there is a natural bijection between \(\mathcal {C}^{k}\) -structures on \(\mathbb {L}\) (up to a \(\mathcal {C}^{k}\) -diffeomorphism fixing both origins 0 and \(\bar{0}\) ) and double D-coset classes \(D\,\backslash \,H\,/\,D = \{ D h D \mid h \in H\}\) . Moreover, the set of all \(\mathcal {C}^{k}\) -structures on \(\mathbb {L}\) (up to a \(\mathcal {C}^{k}\) -diffeomorphism which may also exchange origins) are in one-to-one correspondence with the set of double \((D,\pm )\) -coset classes \(D\,\backslash \,H^{\pm }\,/\,D = \{ D h D \cup D h^{-1} D \mid h \in H\}\) . In particular, in contrast to the real line \(\mathbb {R}\) , the line with two origins \(\mathbb {L}\) admits uncountably many pair-wise non-diffeomorphic \(\mathcal {C}^{k}\) -structures for each \(k=1, 2,\ldots ,\infty \) .