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In the tight-binding description of electronic, photonic, or cold atomic dynamics in a periodic lattice potential, particle motion is described in terms of hopping amplitudes and potentials on an abstract network of discrete sites corresponding to physical orbitals in the lattice. The physical attributes of the orbitals, including their locations in three-dimensional space, are independent pieces of information. In this paper we identify a notion of geometry-independence: any physical quantity or observable that depends only on the tight-binding parameters (and not on the explicit information about the orbital geometry) is said to be "geometry-independent." The band structure itself, and for example the Chern numbers of the bands in a two-dimensional system, are geometry-independent, while the Bloch-band Berry curvature is geometry-dependent. Careful identification of geometry-dependent versus independent quantities can be used as a novel principle for constraining a variety of results. By extending the notion of geometry-independence to certain classes of interacting systems, where the many-body energy gap is evidently geometry-independent, we shed new light on a hypothesized relation between many-body energy gaps of fractional Chern insulators and the uniformity of Bloch band Berry curvature in the Brillouin zone. We furthermore explore the geometry-dependence of semiclassical wave packet dynamics, and use this principle to show how two different types of Hall response measurements may give markedly different results due to the fact that one is geometry-dependent, while the other is geometry-independent. Similar considerations apply for anomalous thermal Hall response, in both electronic and spin systems.