For a field \(F\) of arbitrary characteristic the author introduces the notion of rigidity of a subset \(T\) of \(F^{*2}\) containing \(-1\) relative to an elliptic curve \(X\) with points of order two \(F\)-rational. Thus \(X\) is given by \(y^2 = f(x) = (x-e_1)(x-e_2)(x-e_3)\) where \(e_1,e_2,e_3 \in F\). \((T,X)\) is said to be rigid if whenever a \(F\)-rational point \((r,s)\) lies on the quadratic twist \(X_u\) of \(X\) given by \(uy^2 = f(x)\) where \(u \in T\), then necessarily \(r - e_i \in T \cup 0\) for \(i=1,2,3\). In case of characteristic different from two this is motivated by a condition involving the Weil-Châtelet group. The new rigidity compares nicely with the traditional one: if for all \(w \in F^{*2}\) the \((T \cup wT,X)\) is rigid, then \(T \subset F^*\) is rigid. The main point, however, is the fact that the new rigidity guarantees the existence of valuations on \(F\). More precisely, given a curve \(X\) by the equation \(y^2=f(x)\) as above, and given a subgroup \(T\) of \(F^*\) containing \(F^{*2}\) and such that \((T \cup wT,X)\) is rigid for all \(w \in F^*\), there exists a \(T-\)compatible valuation \(v:F \to \Gamma\) on \(F\) inducing a surjection \(\widetilde{v} : \Gamma/2\Gamma \to F^*/T\). As an application the author discusses the relation between the Witt group WX and the Weil-Châtelet group \(WC(X)\) and also gives an application to the theory of formally real fields.