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Recent numerical simulations of strongly interacting staggered lattice fermions reveal new phases where the fermions can acquire masses without breaking symmetries. I will argue that these results can be understood as a consequence of the cancellation of a new type of anomaly arising from a discrete symmetry. I will show how this discrete symmetry arises in the continuum in theories of Kaehler-Dirac fermions. Remarkably both this symmetry and its associated anomaly survive discretization and restrict the fermion content of lattice theories which are capable of mass generation without symmetry breaking. These results are consistent with results from topological insulators and a new understanding of anomalies rooted in the Dai-Freed theorem.