One way to create new fractals from old fractals is to identify points. We use spectral decimation to investigate the spectrum of the Laplacian on the projective hexagasket, which is obtained from the hexagasket by identifying antipodal points on the outer boundary and on each of a countable set of inner boundaries. We show that the eigenvalue counting function has a power law asymptotics and that the Weyl ratio is asymptotically multiplicatively periodic. We also show that the set of ratios of eigenvalues has gaps. This has interesting consequences in the existence of differential operators on products of the projective hexagasket that are not formally elliptic as differential operators, but in fact are elliptic as pseudodifferential operators.