This claim comes from Henri Darmon, and I would believe it, but can't exactly prove it. Let $\mathcal{F}$ denote the space of $\mathbb{C}$-valued functions on $\mathbb{P}_1(\mathbb{Q})$, and let $\mathcal{M}$ denote the space of $\mathbb{C}$-valued modular symbols. Let $d:\mathcal{F}\to\mathcal{M}$ by $f\mapsto df$ where $(df)\{x\to y\} = f(y) - f(x)$.
The claim is that $d$ is surjective. I see that the right hand side indeed gives a complex valued modular symbol. Is there a very easy way to see that all complex valued modular symbols arise in such a way? Is it obvious?