A large population of combustion-generated soot aggregates (more than 3,000 samples) was thermophoretically extracted from a variety of laminar and turbulent flames and analyzed using transmission electron microscopy (TEM). It was shown that the scaling structural properties of these fractal aggregates cannot be exclusively characterized by a single mass fractal dimension. Asymmetric properties of the aggregates were considered here by first assuming and then demonstrating their self-affinity via. an affinity exponent reflecting scaling with respect to the length and width of the aggregate projections. In addition to the conventional fractal dimension, Df, determined by using the geometrical mean of the longitudinal and transverse sizes as the characteristic length, the affinity exponent, H, and two complementary fractal dimensions, one longitudinal, DL =[(1+H)/2 ]Df, and one transverse, DW =[(1+H)/2H]Df, were introduced. By fitting TEM data for the entire population of aggregates, the values Df = 1.75 and H = 0.91 were obtained. This more complete description of aggregate morphologies in terms of the self-affine scaling is expected to lead to a better understanding of the transport properties and restructuring kinetics of flame-generated aggregates.