We present scattering and rheology studies on model polyelectrolyte complex (PEC) hydrogels that form upon self-assembly of symmetric oppositely charged triblock polyelectrolytes in aqueous media. The hydrogel assembly is driven by associative phase separation of charged end-blocks to form PECs while the neutral mid-blocks restrict bulk phase separation of the PECs, leading to three-dimensional networks with PEC domains surrounded by neutral polymer coronae at sufficiently high polymer concentrations. Comprehensive characterization of the hydrogel structure (PEC domain size, morphology, spacing, and ordering) was enabled by a series of triblock polyelectrolytes with independently varying block lengths, facilitating the construction of morphology maps that offered direct comparisons with earlier theoretical predictions and highlighted the contributions of the charged and neutral blocks in directing PEC morphologies and hydrogel microstructures. Comparisons between the microstructure and shear rheology response emphasize the interplay between PEC domain morphologies and hydrogel flow behaviors. Furthermore, by drawing parallels with theological models for associating polymer networks, we present preliminary insights into the role of chain-exchange dynamics between the PEC domains and cooperative electrostatic interactions in influencing the hydrogel flow properties.