Hamiltonian stationary Lagrangian submanifolds are solutions of a natural and important variational problem in Kaehler geometry. In the particular case of surfaces in Euclidean 4-space, it has recently been proved that the Euler-Lagrange equation is a completely integrable system, which theory allows us to describe all such tori. This article determines the spectral data for these, in terms of a complete algebraic curve, a rational function and a line bundle. We use this data to give explicit formulas for all weakly conformal HSL immersions of a 2-torus into Euclidean 4-space and describe the moduli space of those with given conformal type and Maslov class. We also show that each such torus admits a family of Hamiltonian deformations through HSL tori, the dimension of this family being related to the genus of its spectral curve.