Two coupled resonance modes can lead to exotic transmission spectra due to internal interference processes. Examples include electromagnetically induced transparency (EIT) in atoms and mode splitting in optics. The ability to control individual modes plays a crucial role in controlling such transmission spectra for practical applications. Here we experimentally demonstrate a controllable EIT-like mode splitting in a single microcavity using a double-port excitation. The mode splitting caused by internal coupling between two counter-propagating resonances can be effectively controlled by varying the power of the two inputs, as well as their relative phase. Moreover, the presence of asymmetric scattering in the microcavity leads to chiral behaviors in the mode splitting in the two propagating directions, manifesting itself in terms of a Fano-like resonance mode. These results may offer a compact platform for a tunable device in all-optical information processing.