The exploitable properties of many materials are intimately linked to symmetry-lowering structural phase transitions. We present an automated and exhaustive symmetry-mode method for systematically exploring and solving such structures which will be widely applicable to a range of functional materials. We exemplify the method with an investigation of the Bi2Sn2O7 pyrochlore, which has been shown to undergo transitions from a parent gamma cubic phase to beta and alpha structures on cooling. The results include the first reliable structural model for beta-Bi2Sn2O7 (orthorhombic Aba2, a = 7.571833(8), b = 21.41262(2), and c = 15.132459(14) angstrom) and a much simpler description of alpha-Bi2Sn2O7 (monoclinic Cc, a = 13.15493(6), b = 7.54118(4), and c = 15.07672(7) angstrom, beta = 125.0120(3)degrees) than has been presented previously. We use the symmetry-mode basis to describe the phase transition in terms of coupled rotations of the Bi2O' anti-cristobalite framework, which allow Bi atoms to adopt low-symmetry coordination environments favored by lone-pair cations.