We discovered a new transmodal Fabry-Pérot resonance where one elastic-wave mode is maximally transmitted to another. It occurs when the phase difference of two dissimilar modes through an anisotropic layer becomes odd multiples of π under the reflection-free and weak mode-coupling assumptions. Unlike the well-established Fabry-Pérot resonance, the transmodal resonance must involve two coupled elastic waves between longitudinal and shear modes. The investigation into the origin of wiggly transmodal transmission spectra suggests that efficient broadband mode conversion can be achieved if the media satisfy the structural stability condition to some degree. The new resonance mechanism, also experimentally characterized, opens up new possibilities for manipulating elastic wave modes as an effective alternative to generating shear-mode ultrasound.