Let θ be an inner function satisfying the connected level set condition of B. Cohn, and let Kθ1 be the shift-coinvariant subspace of the Hardy space H¹ generated by θ. We describe the dual space to Kθ1 in terms of a bounded mean oscillation with respect to the Clark measure σ_α of θ. Namely, we prove that (Kθ1∩zH1)*=BMO(σα). The result yields a two-sided estimate for the operator norm of a finite Hankel matrix of size n×n via BMO(μ_2n)-norm of its standard symbol, where μ_2n is the Haar measure on the group {ξ∈C:ξ2n=1}.