  We identify some hidden symmetries of Chern-Simons theories, such as appear
in the effective theory for quantized Hall states. This allows us to determine
which filling fractions admit spin-singlet quantum Hall states. Our results
shed some light on states already observed at $\nu=2/3$, and transitions
between them. We identify SU(2), or higher, symmetries of many additional
states -- including spin-polarized states. Our symmetries classify low-lying
excited states and may be of use in the construction of trial wavefunctions,
but are typically not present in the edge theory, where they are lifted by
non-universal couplings.
