We investigate the relationship between the complex symmetry of composition operators Cϕf=f∘ϕ induced on the classical Hardy space H2(D) by an analytic self-map ϕ of the open unit disk D and its Koenigs eigenfunction. A generalization of orthogonality known as conjugate-orthogonality will play a key role in this work. We show that if ϕ is a Schröder map (fixes a point a∈D with 0<|ϕ′(a)|<1) and σ is its Koenigs eigenfunction, then Cϕ is complex symmetric if and only if (σn)n∈N is complete and conjugate-orthogonal in H2(D). We study the conjugate-orthogonality of Koenigs sequences with some concrete examples. We use these results to show that commutants of complex symmetric composition operators with Schröder symbols consist entirely of complex symmetric operators.