Discrete degrees of freedom, such as spin and orbital, provide a tool to manipulate electrons, photons and phonons. Topological insulators have stimulated intense interests in condensed-matter physics, optics, acoustics and mechanics, usually with a focus on the spin degree of freedom. However, the orbital degree of freedom constitutes another fundamental attribute in crystals, but has seldom been investigated in topological insulators. Here, we demonstrate topological insulators with an auxiliary orbital degree of freedom on a nanomechanical platform. We realize an adiabatic transition between distinct topological edge states, which constitutes a crucial functionality for integrated circuits accommodating distinct topological edge channels. Beyond the one-dimensional edge states, we further construct zero-dimensional Dirac-vortex states using the orbital degree of freedom. These nanomechanical Dirac-vortex states exhibit strong second-order and third-order nonlinearities. Our results introduce the orbital degree of freedom as an alternative means to manipulate the topological phase transition on an integrated platform.