In this work we study timelike circular orbits and photon orbits at the throat of stationary and axisymmetric wormholes. We assume a global radial coordinate l connecting both sides, with twice differentiable metric components and vanishing first derivatives at the throat. With regard to the specific angular momentum ℓ of a test particle, we derive expressions for the boundaries of a spectrum of bound circular orbits. We further express these boundaries in terms of physical properties of the spacetime. With regard to the parameter space behavior, we found a phase transition occurring for wormholes with an ergoregion. Furthermore, the presented expressions could give constraints on wormhole properties, as they hint to possible instabilities, especially for fast rotating wormholes and wormholes with an ergoregion.