  Expressions for neutrino oscillations contain a high degree of symmetry, but
typical forms for the oscillation probabilities mask these symmetries of the
oscillation parameters. We elucidate the $2^7$ parameter symmetries of the
vacuum parameters and draw connections to the choice of definitions of the
parameters as well as interesting degeneracies. We also show that in the
presence of matter an \emph{additional} set of $2^7$ parameter symmetries exist
of the matter parameters. Due to the complexity of the exact expressions for
neutrino oscillations in matter, numerous approximations have been developed;
we show that under certain assumptions, approximate expressions have at most
$2^6$ additional parameter symmetries of the matter parameters. We also include
one parameter symmetry related to the LMA-Dark degeneracy that holds under the
assumption of CPT invariance; this adds one additional factor of two to all of
the above cases. Explicit, non-trivial examples are given of how physical
observables in neutrino oscillations, such as the probabilities, CP violation,
the position of the solar and atmospheric resonance, and the effective $\Delta
m^2$'s for disappearance probabilities, are invariant under all of the above
symmetries. We investigate which of these parameter symmetries apply to
numerous approximate expressions in the literature and show that a more careful
consideration of symmetries improves the precision of approximations.
