Let (X, d) and \((Y,\rho )\) be compact metric spaces and let \(\tau \) and \(\eta \) respectively be Lipschitz involutions on (X, d) and \((Y,\rho )\) . In this paper, we first give a complete description of surjective real linear uniform isometries between \(\textrm{Lip}(X,d,\tau )\) and \(\textrm{Lip}(Y,\rho ,\eta )\) . We next study 2-local real uniform isometries from \(\textrm{Lip}(X,d,\tau )\) to \(\textrm{Lip}(Y,\rho ,\eta )\) . For such a map T, we show that \(|T(1_{X})|=1_{Y}\) on Y and there exist a nonempty \(\eta \) -invariant subset \(Y_{0}\) of Y which is a boundary for \(T(\textrm{Lip}(X,d,\tau ))\) with respect to Y and a bijective continuous map \(\varphi : Y_{0}\rightarrow X\) with \(\varphi \circ \eta =\tau \circ \varphi \) on \(Y_{0}\) such that \(T(f)(y)=T(1_{X})(y)f(\varphi (y))\) for all \(f\in \textrm{Lip}(X,d,\tau )\) and \(y\in Y_{0}\) . In continuation, we prove that \(Y_{0}=Y\) and \(\varphi \) is a Lipschitz mapping from \((Y,\rho )\) to (X, d) whenever \(\tau \) is an isometric involution on (X, d). In particular, we show that every 2-local real uniform isometry from \(\textrm{Lip}_{\mathbb {R}}(X,d)\) to \(\textrm{Lip}_{\mathbb {R}}(Y,\rho )\) is a surjective real linear uniform isometry.
