Best-fit values of recent global analyses of neutrino data imply large solar neutrino mixing, vanishing $U_{e3}$, and a nonmaximal atmospheric neutrino mixing angle ${\theta }_{23}$. We show that these values emerge naturally by the hypothesis of scaling in the Majorana neutrino mass matrix, which states that the ratios of its elements are equal. It also predicts an inverted hierarchy for the neutrino masses. We point out several advantages and distinguishing tests of the scaling hypothesis compared to the $L_e-L_{\mu }-L_{\tau }$ flavor symmetry, which is usually assumed to provide an understanding of the inverted hierarchy. Scenarios which have initially vanishing $U_{e3}$ and maximal atmospheric neutrino mixing are shown to be unlikely to lead to nonmaximal ${\theta }_{23}$ while simultaneously keeping $U_{e3}$ zero. We find a peculiar ratio of the branching ratios $\mu \to e\gamma$ and $\tau \to e\gamma$ in supersymmetric seesaw frameworks, which only depends on atmospheric neutrino mixing and results in $\tau \to e\gamma$ being unobservable. The consequences of the scaling hypothesis for high energy astrophysical neutrinos at neutrino telescopes are also investigated. Then we analyze a seesaw model based on the discrete symmetry $D_4\times Z_2$ leading to scaling in the low energy mass matrix and being capable of generating the baryon asymmetry of the Universe via leptogenesis. The relevant $CP$ phase is identical to the low energy Majorana phase, and successful leptogenesis requires an effective mass for neutrinoless double beta decay larger than 0.045 eV.

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