We have patterned novel Permalloy thin films with quasicrystalline Penrose P2 tilings and measured their dc magnetization and ferromagnetic resonance absorption. Reproducible anomalies in the hysteretic, low-field data signal a series of abrupt transitions between ordered magnetization textures, culminating in a smooth evolution into a saturated state. Micromagnetic simulations compare well to experimental dc hysteresis loops and ferromagnetic resonance spectra and indicate that systematic control of magnetic reversal and domain wall motion can be achieved via tiling design, offering a new paradigm of magnonic quasicrystals.