The paper deals with a Mehler representation for the Jacobi functions \(\varphi_\lambda^{(\alpha,\beta)}(t)\) with respect to the dual variable \(\lambda\). This representation is used to define a pair of dual transforms of Mehler type: \(\chi_{\alpha,\beta}\) and its transposed \(^t\chi_{\alpha,\beta}\). Two second order difference operators \(P_{\alpha,\beta}\) and \(Q\) are studied, such that the Jacobi function \(\varphi_\lambda^{(\alpha,\beta)}(t)\) is an eigenfunction of \(P_{\alpha,\beta}\) with respect to the dual variable, and \(\chi_{\alpha,\beta}\) and its transposed \(^t\chi_{\alpha,\beta}\) are permutation operators between \(P_{\alpha,\beta}\) and \(Q\). The authors study some spaces of functions on which both transforms are isomorphisms and they find inversion formulae for these transforms.