Infrared nanoimaging of phonon polaritons in twisted alpha-phase molybdenum trioxide bilayers reveals tunable wavefront geometries and topological transitions over a broad range of twist angles, offering a configurable platform for nanophotonic applications. Moire engineering is being intensively investigated as a method to tune the electronic, magnetic and optical properties of twisted van der Waals materials. Advances in moire engineering stem from the formation of peculiar moire superlattices at small, specific twist angles. Here we report configurable nanoscale light-matter waves-phonon polaritons-by twisting stacked alpha-phase molybdenum trioxide (alpha-MoO3) slabs over a broad range of twist angles from 0 degrees to 90 degrees. Our combined experimental and theoretical results reveal a variety of polariton wavefront geometries and topological transitions as a function of the twist angle. In contrast to the origin of the modified electronic band structure in moire superlattices, the polariton twisting configuration is attributed to the electromagnetic interaction of highly anisotropic hyperbolic polaritons in stacked alpha-MoO(3)slabs. These results indicate twisted alpha-MoO(3)to be a promising platform for nanophotonic devices with tunable functionalities.