A tire changer machine, and method of operation, configured with a tire-engaging wheel clamp assembly to secure a wheel assembly consisting of a wheel rim and tire mounted thereon to a drive assembly, and for demounting of the tire from the wheel rim. Upon dismounting of the tire from the wheel rim, a recessed surface and annular flange of the tire-engaging wheel clamp assembly is configured to catch and retain a rim or bead of the demounted tire, restricting movement of the tire upon dismounting from the wheel rim, until retrieved by an operator.