What is the mechanism for this Horner-Wadsworth-Emmons (HWE) reaction?
First dil. aq. HCl hydrolyses the starting material to the dialdehyde.
Second the diethylphosphonoacetate anion reacts with one of the aldehyde groups to give the expected intermediate alkoxide. This is in equilibrium with the carbanion shown below. This is not normally depicted in the HWE mechanism as there is usually no pathway forward from it, however in this case there is. This carbanion can, and does, undergo a second HWE with the other aldehyde which proceeds through the usual mechanism to give the products shown in the question