Acoustic spin textures provide a degree of freedom for programmable topological field states, yet a unified framework for constructing and classifying higher-order textures in open-air acoustics remains lacking. Here we introduce polygonal standing-wave interference as a framework connecting discrete symmetry, angular phase encoding, phase singularities, chiral time-averaged energy flux, and acoustic spin. By controlling the number and relative phases of the standing-wave channels, we experimentally realize D4 meron/anti-meron lattice, D6 double-skyrmion superlattice, and D8 acoustic-spin quasicrystal composed of four meron/anti-meron sublattices. Finite polygonal fields are normalized angular discretizations of continuous standing-wave spectra. Under the m=2 phase encoding, the D6 and D8 fields reproduce the local q=+2 core of a second-order Bessel vortex while retaining distinct long-range order through their discrete interference channels. At fixed m increasing N under normalized angular sampling yields the continuous Bessel limit. These results establish phase encoding, pairwise interference, reciprocal-space composition, and angular discretization as a unified design principle for programmable topological acoustic spin textures.