Armchair graphene nanoribbons are a highly promising class of semiconductors for all-carbon nanocircuitry. Here, we present a new perspective on their electronic structure from simple model Hamiltonians and ab initio calculations. We focus on a specific set of nanoribbons of width n=3p+2, where n is the number of carbon atoms across the nanoribbon axis and p is a positive integer. We demonstrate that the energy-gap opening in these nanoribbons originates from the breaking of a previously unidentified hidden symmetry by long-ranged hopping of π electrons and structural distortions occurring at the edges. This hidden symmetry can be restored or manipulated through the application of in-plane lattice strain, which enables continuous energy-gap tuning, the emergence of Dirac points at the Fermi level, and topological quantum phase transitions. Our work establishes an original interpretation of the semiconducting character of armchair graphene nanoribbons and offers guidelines for rationally designing their electronic structure.