Knutson-Tao puzzles are fun combinatorial gadgets that perform Schubert calculus -- they compute the structure constants of the cohomology ring of the Grassmannian. Another way to compute these same constants is by counting the points in triple intersections of Schubert varieties. Thus Schubert calculus bridges the worlds of combinatorics and geometry. Symmetries that are plainly visible in one context may be non-obvious in the other, in which case we learn something new. As an example of this interplay, 180 degree rotation of a parallelogram-shaped puzzle easily reveals a geometric symmetry, whose geometric proof in turn uncovers an even stronger symmetry of parallelogram-shaped puzzles.