The article concerns Weingarten surfaces defined by the equation \[ K-2mH+ m^2-l^2=0 \] with the constants \(m,l\) satisfying \(m^2+l^2>0\). The fundamental forms can be adapted as \[ \begin{aligned} I&= \biggl(\cos^2 \frac\varphi 2 du^2+\sin^2 \frac\varphi 2 dv^2\biggr)/ (m^2+l^2),\\ II&= \biggl(\sin \frac {\varphi-\varphi_0}{2} \cos \frac \varphi 2 du^2-\cos \frac {\varphi-\varphi_0}{2} \sin\frac \varphi 2 dv^2\biggr)/ (m^2+l^2)^{1/2}, \end{aligned} \] with the Chebyshev angle \(\varphi\) satisfying the sine-Gordon equation \(\varphi_{uu}- \varphi_{vv}= \sin\varphi\). Let \(f:S\to s^*\) be a correspondence between two surfaces such that (i) the distance between \(P\in S\) and \(f(p)\in S^*\) is constant, (ii) the angles between the line \(\overline{Pf(P)}\) and surfaces \(S\), \(S^*\) are constant, (iii) the angles between the normals at \(P\in S\) and \(f(P)\in S^*\) are constant. Then \(S\), \(S^*\) are Weingarten surfaces of the above kind, and the correspondence \(f\) realizes the classical Bäcklund transformation between the relevant sine-Gordon equations relevant to \(S\) and \(S^*\).