We propose that three-dimensional (3D) layered octagonal quasicrystals can host magnetic-field tunable Möbius insulators and various higher-order topological insulators (HOTIs), enabled by the interplay of quasicrystalline symmetry and magnetic order. By constructing a minimal model based on stacked Ammann-Beenker tilings with magnetic exchange coupling and octagonal warping, we demonstrate that an A-type antiferromagnetic (AFM) configuration yields a topological phase protected by an effective time-reversal symmetry $\mathcal{S}=\mathcal{T}{\tau }_{1/2}$. Breaking $\mathcal{S}$ via an in-plane magnetic field induced canting of the AFM order while preserving a nonsymmorphic glide symmetry ${\mathcal{G}}_n={\tau }_{1/2}{\mathcal{M}}_n$ leads to Möbius-twisted surface states, realizing a Möbius insulator in an aperiodic 3D system. Furthermore, we show that the quasicrystal with a general magnetic configuration supports multiple HOTI phases characterized by distinct hinge mode configurations that can be switched by rotating the magnetic field. A low-energy effective theory reveals that these transitions are driven by mass kinks between adjacent surfaces. Our work establishes a platform for realizing symmetry-protected topological phases unique to quasicrystals and highlights the tunability of hinge and surface states via magnetic control.

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• Magnetic order
• Topological phases of matter
• Quasicrystals
• Tight-binding model