This paper reports the computational simulation of the correlations between atomic-level stress and local structure fluctuations in a computational model of amorphous silicon. A single parameter has been identified, which uniquely characterises the structural order in these structures. This parameter is the linear combination of the SDs of the first and second nearest neighbour separations. The stress fluctuations, under progressive hydrogen incorporation, show two clear dependences on this parameter, and therefore on the structural order. This dual dependency clearly alludes to structural network configurations that contain low and high hydrogen concentrations. The implications of the results on the local geometry of tetrahedrally bonded amorphous solids are discussed.