  Using an algebraic formulation, we explore two well-known degeneracies
involving the mass-squared differences for three-neutrino oscillations assuming
CP symmetry is conserved. For vacuum oscillation, we derive the expression for
the mixing angles that permit invariance under the interchange of two
mass-squared differences. This symmetry is most easily expressed in terms of an
ascending mass order. This can be used to reduce the parameter space by one
half in the absence of the MSW effect. For oscillations in matter, we derive
within our formalism the known approximate degeneracy between the standard and
inverted mass hierarchies in the limit of vanishing $\theta_{13}$. This is done
with a mass ordering that permits the map $\Delta_{31} \mapsto -\Delta_{31}$.
Our techniques allow us to translate mixing angles in this mass order
convention into their values for the ascending order convention. Using this
dictionary, we demonstrate that the vacuum symmetry and the approximate
symmetry invoked for oscillations in matter are distinctly different.
