How can children's natural perceptuo-motor skills be harnessed for teaching and learning mathematical structure? We address this question in the case of the integers. Existing research suggests that adult mental representations of integers recruit perceptuo-motor functionalities involving symmetry. Building on these findings, we designed a hands-on curriculum that emphasizes symmetry to teach integer concepts to fourth graders. Compared to two control conditions, children who went through the experimental curriculum showed evidence of incorporating symmetry into their mental representations of integers and performed higher on problems beyond the scope of instruction, including negative fractions and algebra-readiness problems. Gains did not come at the expense of basic integer computation skill. This study has direct practical implications, as current integers curricula generally omit symmetry. The research demonstrates an approach to designing instruction that involves identifying perceptuo-motor functionalities underlying numerical cognition and creating learning activities to recruit them.