We examine the phase response properties of half-center oscillators (HCOs) that are modeled by a pair of Morris-Lecar-type neurons connected by strong fast inhibitory synapses. We find that the two basic mechanisms for half-center oscillations, "release" and "escape", give rise to strikingly different phase response curves (PRCs). Release-type HCOs are most sensitive to perturbations delivered to cells at times when they are about to transition from the active to the suppressed state, and PRCs are dominated by a large negative peak (phase delays) at corresponding phases. On the other hand, escape-type HCOs are most sensitive to perturbations delivered to cells at times when they are about to transition from the suppressed to the active state, and PRCs are dominated by a large positive peak (phase advances) at corresponding phases. By analyzing the phase space structure of Morris-Lecar-type HCO models with fast synaptic dynamics, we identify the dynamical mechanisms underlying the shapes of the PRCs. To demonstrate the significance of the different shapes of the PRCs for the release-type and escape-type HCOs, we link the shapes of the PRCs to the different frequency modulation properties of release-type and escape-type HCOs, and we show that the different shapes of the PRCs for the release-type and escape-type HCOs can lead to fundamentally different phase-locking dynamics.