This paper presents a phenomenological theory of alternagnets, a new class of magnetic materials with unique spin textures and dynamics. The theory explains the characteristic lifted degeneracy of magnon spectra in d-wave alternagnets like RuO₂ by introducing an effective sublattice-dependent anisotropic spin stiffness. This stiffness arises naturally from the theory and leads to finite magnetization gradients in alternagnetic domain walls, even for 180° domain walls. These gradients generate a ponderomotive force in the presence of a strongly inhomogeneous external magnetic field, which can be measured using magnetic force microscopy. The motion of alternagnetic domain walls is characterized by anisotropic Walker breakdown, with much higher speed limits than ferromagnets but lower than antiferromagnets. Alternagnets exhibit spin-polarized d/g/i-wave order in the nonrelativistic band structure, distinct from conventional ferromagnets and antiferromagnets. They combine the fast magnetic dynamics and robustness to external fields of antiferromagnets with the strong spin-dependent splitting of electronic bands typical of ferromagnets, offering new functionalities in spintronic applications. Alternagnets also showcase unique phenomena such as the crystal anomalous Hall effect and the spin splitter effect, distinct spectroscopic signatures, and magnonic spectra with anisotropic lifted degeneracy. The properties of alternagnets arise from their defining spin symmetries, which enforce magnetic compensation and time-reversal symmetry breaking in the non-relativistic band structure. The theory confirms the predicted non-relativistic spin splitting of the magnon spectra and predicts several unique properties of alternagnetic textures, including inhomogeneous magnetization distribution inside domain walls, the possibility to manipulate domain walls with magnetic force microscopy tools, and anisotropic Walker breakdown in domain wall motion. These features are explained by the emergence of an effective sublattice-dependent anisotropic spin stiffness, whose symmetry corresponds to the alternagnetic spin splitting of electronic bands. The results are supported by spin lattice model simulations. The paper also discusses the deformation of moving domain walls and the Walker breakdown, showing that alternagnetic domain walls can be controlled by inhomogeneous magnetic fields and their velocity is constrained by the Walker breakdown, similar to ferromagnets. The anisotropic splitting of magnon bands represents distinctive magnetic characteristics, and these effects can occur in insulators, making them efficient tools for identifying alternative materials. The study highlights the unique properties of alternagnets and their potential applications in spintronics and magnetic materials research.This paper presents a phenomenological theory of alternagnets, a new class of magnetic materials with unique spin textures and dynamics. The theory explains the characteristic lifted degeneracy of magnon spectra in d-wave alternagnets like RuO₂ by introducing an effective sublattice-dependent anisotropic spin stiffness. This stiffness arises naturally from the theory and leads to finite magnetization gradients in alternagnetic domain walls, even for 180° domain walls. These gradients generate a ponderomotive force in the presence of a strongly inhomogeneous external magnetic field, which can be measured using magnetic force microscopy. The motion of alternagnetic domain walls is characterized by anisotropic Walker breakdown, with much higher speed limits than ferromagnets but lower than antiferromagnets. Alternagnets exhibit spin-polarized d/g/i-wave order in the nonrelativistic band structure, distinct from conventional ferromagnets and antiferromagnets. They combine the fast magnetic dynamics and robustness to external fields of antiferromagnets with the strong spin-dependent splitting of electronic bands typical of ferromagnets, offering new functionalities in spintronic applications. Alternagnets also showcase unique phenomena such as the crystal anomalous Hall effect and the spin splitter effect, distinct spectroscopic signatures, and magnonic spectra with anisotropic lifted degeneracy. The properties of alternagnets arise from their defining spin symmetries, which enforce magnetic compensation and time-reversal symmetry breaking in the non-relativistic band structure. The theory confirms the predicted non-relativistic spin splitting of the magnon spectra and predicts several unique properties of alternagnetic textures, including inhomogeneous magnetization distribution inside domain walls, the possibility to manipulate domain walls with magnetic force microscopy tools, and anisotropic Walker breakdown in domain wall motion. These features are explained by the emergence of an effective sublattice-dependent anisotropic spin stiffness, whose symmetry corresponds to the alternagnetic spin splitting of electronic bands. The results are supported by spin lattice model simulations. The paper also discusses the deformation of moving domain walls and the Walker breakdown, showing that alternagnetic domain walls can be controlled by inhomogeneous magnetic fields and their velocity is constrained by the Walker breakdown, similar to ferromagnets. The anisotropic splitting of magnon bands represents distinctive magnetic characteristics, and these effects can occur in insulators, making them efficient tools for identifying alternative materials. The study highlights the unique properties of alternagnets and their potential applications in spintronics and magnetic materials research.