Symmetry in the sequence of approximation coefficients

Let {an}^∞₁ and {θn}^∞₀ be the sequences of partial quotients and approximation coefficients for the continued fraction expansion of an irrational number. We will provide a function f such that a_n+1 = f(θ_n±1, θ_n). In tandem with a formula due to Dajani and Kraaikamp, we will write θ_n±1 as a function of (θ_n∓1, θn), revealing an elegant symmetry in this classical sequence and allowing for its recovery from a pair of consecutive terms.