The incorporation of nonhexagonal rings into graphene nanoribbons (GNRs) is an effective strategy for engineering localized electronic states, bandgaps, and magnetic properties. Here, we demonstrate the successful synthesis of nanoribbons having four-membered ring (cyclobutadienoid) linkages by using an on-surface synthesis approach involving direct contact transfer of coronene-type precursors followed by thermally assisted [2 + 2] cycloaddition. The resulting coronene-cyclobutadienoid nanoribbons feature a narrow 600-meV bandgap and novel electronic frontier states that can be interpreted as linear chains of effective p_x and p_y pseudo-atomic orbitals. We show that these states give rise to exceptional physical properties, such as a rigid indirect energy gap. This provides a previously unexplored strategy for constructing narrow gap GNRs via modification of precursor molecules whose function is to modulate the coupling between adjacent four-membered ring states.