Bounded domains have discrete eigenfrequencies/spectra, and cavities with different boundaries and areas have different spectra. A general methodology for isospectral twinning, whereby the spectra of different cavities are made to coincide, is created by combining ideas from across physics including transformation optics, inverse problems, and metamaterial cloaking. We twin a hexagonal drum with a deformed hexagonal drum using a nonsingular coordinate transform that adjusts the deformed shape by mapping a near boundary domain to a zone of heterogeneous anisotropic medium. Splines define the mapping zone for twinning these two drums and we verify isospectrality by a finite-element analysis.