In this work, bowtie nanoantennas and nanocavities are analyzed using the conformal transformation technique. Their performance is studied in terms of the non-radiative Purcell enhancement and self-induced optical forces experienced by quantum emitters. It is demonstrated how these two geometrically different plasmonic nanoparticles can share the same non-radiative Purcell spectra. This hidden symmetric response is unveiled by properly applying the conformal transformation technique, demonstrating that both nanoparticles share the same transformed geometry.