The critically important at-a-station hydraulic geometry (AHG) relationships relate hydraulic variables (depth, width, or velocity) to discharge in power law form. The recently discovered at-many-stations hydraulic geometry (AMHG) states that AHG exponents and coefficients are strongly correlated, removing one parameter and lending AMHGs to remote sensing discharge estimation. Despite the excitement, there is an urgent need to clarify AMHG's geomorphological significance for different hydraulic variables. Using data from 57 rivers in the U.S. and perturbation experiments, we show that the width-AMHG is weak, arising mainly from the mathematical construct involving the exponent in both the regressor and the regressand. In contrast, the depth-AMHGs result from geomorphological coevolution. The similar-time-mean condition, as argued for in the literature, is sufficient but not necessary for rating curve convergence and in turn AMHG, for depth and velocity. The predictive accuracy of AMHGs, whose coefficient of determination is unit dependent, is similar to the flow percentile-based downstream hydraulic geometry.