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Altermagnets break time-reversal symmetry, and their spin-orbit coupling (SOC) allows for an anomalous Hall effect (AHE) that depends on the direction of the N´eel ordering vector. The AHE and the ferromagnetic spin moment share the same symmetry and hence are usually proportional. However, density functional theory (DFT) calculations find that the AHE exists with negligible ferromagnetic spin moment for some compounds, whereas it reaches sizable values for other altermagnets. By examining realistic minimal models for altermagnetism in which the DFT phenomenology is captured, we uncover a general SOC-enabled quasisymmetry, the uniaxial spin space group, that provides a natural explanation for the amplitude of the ferromagnetic spin moment across the vast range of different altermagnetic materials. Additionally, we derive analytic expressions for the magnetic anisotropy energy, providing a simple means of identifying the preferred N´eel vector orientation for altermagnets.