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, D
, 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, D
, and one transverse, D
, were introduced. By fitting TEM data for the entire population of aggregates, the values D
= 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.