  Spectral statistics such as the level spacing statistics and spectral form
factor (SFF) are widely expected to accurately identify ``ergodicity'',
including the presence of underlying macroscopic symmetries, in generic quantum
systems ranging from quantized chaotic maps to interacting many-body systems.
By studying various quantizations of maximally chaotic maps that break a
discrete classical symmetry upon quantization, we demonstrate that this
approach can be misleading and fail to detect macroscopic symmetries. Notably,
the same classical map can exhibit signatures of different random matrix
symmetry classes in short-range spectral statistics depending on the
quantization. While the long-range spectral statistics encoded in the early
time ramp of the SFF are more robust and correctly identify macroscopic
symmetries in several common quantizations, we also demonstrate analytically
and numerically that the presence of Berry-like phases in the quantization
leads to spectral anomalies, which break this correspondence. Finally, we
provide numerical evidence that long-range spectral rigidity remains directly
correlated with ergodicity in the quantum dynamical sense of visiting a
complete orthonormal basis.
