A large population of combustion-generated soot aggregates (more than 3000 samples) was thermophoretically extracted from a variety of laminar and turbulent flames and analyzed by 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 affinity exponents reflecting different scaling with respect to the length and width of the aggregate projections. In addition to the conventional fractal dimension, D-f, 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-L = [(1 + H)/2]D-f, and one transverse, D-w = [(1 + H)/2H]D-f, were introduced. By fitting the TEM data for the entire population of aggregates, the values of D-f = 1.75 and H = 0.91 were obtained. To classify the density and crossover scales of aggregates having the same fractal dimensions, lacunarities of the first and second order were also defined as prefactors in the scaling relationships among aggregate mean mass, rms mass and linear sizes. Analysis of the second moment of the mass-size distribution confirmed that the scaling properties of flame-generated aggregates cannot be consummately characterized by a single fractal dimension; it is necessary to introduce a set of scaling exponents. This more precise description of aggregate morphologies in terms of self-affine scaling and lacunarities is not captured by previous idealized cluster-cluster aggregation models. Current investigations of the reasons for this are expected to lead to a deeper understanding of the coagulation dynamics, transport properties, and restructuring kinetics of flame-generated aggregates.