We have used the XAFS (x-ray-absorption fine structure) technique to investigate the local structure about the Cu, Ba, and Tl atoms in orthorhombic ${\mathrm{Tl}}_2$${\mathrm{Ba}}_2$${\mathrm{CuO}}_{\mathit{y}}$ with a superconducting transition temperature ${\mathit{T}}_{\mathit{c}}$=60 K. Our results clearly show that the O(1), O(2), Cu, and Ba atoms are at their ideal sites as given by the diffraction measurements, while the Tl and O(3) atoms are more disordered than suggested by the average crystal structure. The Tl-Tl distance at 3.5 Å between the TlO layers does not change, but the Tl-Tl distance at 3.9 Å within the TlO layer is not observed and the Tl-Ba and Ba-Tl peaks are very broad. The shorter Tl-O(3) distance in the TlO layer is about 2.33 Å, significantly shorter than the distance calculated with both the Tl and O(3) atoms at their ideal 4e sites (x=y=0 or 1/2). A model based on these results shows that the Tl atom is displaced along the 〈110〉 directions from its ideal site by about 0.11 Å; the displacements of neighboring Tl atoms are correlated, with a local antiferroelectriclike order. The O(3) atom is shifted from the 4e site by about 0.53 Å roughly along the 〈100〉 directions. A comparison of the Tl ${\mathit{L}}_{\mathrm{III}}$-edge XAFS spectra from three samples, with ${\mathit{T}}_{\mathit{c}}$=60, 76, and 89 K, shows that the O environment around the Tl atom is sensitive to ${\mathit{T}}_{\mathit{c}}$ while the Tl local displacement is insensitive to ${\mathit{T}}_{\mathit{c}}$ and the structural symmetry. These conclusions are compared with other experimental results and the implications for charge transfer and superconductivity are discussed.

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