We generalize classical triangular Schubert puzzles to puzzles with convex polygonal boundary. We give these puzzles a geometric Schubert calculus interpretation and derive novel combinatorial commutativity statements, using purely geometric arguments, for puzzles with four, five, and six sides, having various types of symmetry in their boundary conditions. We provide alternate combinatorial proofs as well. We also present formulas for the associated structure constants in terms of Littlewood-Richardson numbers, and we prove an analogue of commutativity for parallelogram-shaped equivariant puzzles. We discuss further extensions of the results to puzzles that perform computations in other cohomology theories, such as K-theory, and 2-step and 3-step puzzles.