That is, in fact, true. Planes can fly with symmetrical wings, including flat wings, and even with upside-down wings, as is easily demonstrated by observing stunt planes actually flying upside-down--in all cases, as long as they are appropriately tilted to redirect the air. Most purpose-built stunt planes even actually have symmetrically-curved wings, because the reduction in right-way-up aerodynamic performance is made up for by the increase in inverted performance.