  The recent discovery and realizations of higher-order topological insulators
enrich the fundamental studies on topological phases. Here, we report
three-dimensional (3D) wave-steering capabilities enabled by topological
boundary states at three different orders in a 3D phononic crystal with
nontrivial bulk topology originated from the synergy of mirror symmetry of the
unit cell and a non-symmorphic glide symmetry of the lattice. The multitude of
topological states brings diverse possibility of wave manipulations. Through
judicious engineering of the boundary modes, we experimentally demonstrate two
functionalities at different dimensions: 2D negative refraction of sound wave
enabled by a first-order topological surface state with negative dispersion,
and a 3D acoustic interferometer leveraging on second-order topological hinge
states. Our work showcases that topological modes at different orders promise
diverse wave steering applications across different dimensions.
