This paper presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials, are exactly the 3-point Gromov-Witten invariants, which are the structure constants of the quantum cohomology ring. This construction implies three symmetries of the Gromov-Witten invariants of the Grassmannian with respect to the groups $S_3$, ${\left(\mathbb{Z}/n\mathbb{Z}\right)}^2$, and $\mathbb{Z}/2\mathbb{Z}$. The last symmetry is a certain \emph{curious duality} of the quantum cohomology which inverts the quantum parameter $q$. Our construction gives a solution to a problem posed by Fulton and Woodward about the characterization of the powers of the quantum parameter $q$ which occur with nonzero coefficients in the quantum product of two Schubert classes. The curious duality switches the smallest such power of $q$ with the highest power. We also discuss the affine nil-Temperley-Lieb algebra that gives a model for the quantum cohomology.