Let (M, g) and (M′,g′)\documentclass[12pt]{minimal}
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\begin{document}$$(M',g')$$\end{document} be non-orientable Riemannian surfaces with fixed boundary Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} and fixed Euler characterictic m, and Λ\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda $$\end{document} and Λ′\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda '$$\end{document} be their Dirichlet-to-Neumann maps, respectively. We prove that the closeness of Λ′\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda '$$\end{document} to Λ\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda $$\end{document} in the operator norm implies the existence of the near-conformal diffeomorphism β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} between (M, g) and (M′,g′)\documentclass[12pt]{minimal}
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\begin{document}$$(M',g')$$\end{document} which does not move the points of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}. Thereby we establish the continuity of the determination Λ↦[(M,g)]\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda \mapsto [(M,g)]$$\end{document}, where [(M, g)] is the conformal class of (M, g) and the set of such conformal classes is endowed with the natural Teichmüller-type metric dT\documentclass[12pt]{minimal}
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\begin{document}$$d_T$$\end{document}. In both orientable and non-orientable case we provide quantitative estimates of dT([(M,g)],[(M′,g′)])\documentclass[12pt]{minimal}
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\begin{document}$$d_T([(M,g)],[(M',g')])$$\end{document} via the operator norm of the difference Λ′-Λ\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda '-\Lambda $$\end{document}. We also obtain generalizations of the results above to the case in which the Dirichlet-to-Neumann map is given only on a segment of the boundary.
