It is known that, given a genus 2 curve $${{\mathcal C} : {y^2 = f(x)}}$$ , where f(x) is quintic and defined over a field K, of characteristic different from 2, and given a homogeneous space $${{\mathcal H}_\delta}$$ for complete 2-descent on the Jacobian of $${{\mathcal C}}$$ , there is a V δ (which we shall describe), which is a degree 4 del Pezzo surface defined over K, such that $${{\mathcal H}_\delta(K) \not= \emptyset \implies V_\delta(K) \not= \emptyset}$$ . We shall prove that every degree 4 del Pezzo surface V, defined over K, arises in this way; furthermore, we shall show explicitly how, given V, to find $${{\mathcal C}}$$ and δ such that V = V δ , up to a linear change in variable defined over K. We shall also apply this relationship to Hürlimann’s example of a degree 4 del Pezzo surface violating the Hasse principle, and derive an explicit parametrised infinite family of genus 2 curves, defined over $${{\mathbb Q}}$$ , whose Jacobians have nontrivial members of the Shafarevich-Tate group. This example will differ from previous examples in the literature by having only two $${{\mathbb Q}}$$ -rational Weierstrass points.