Cross-modality recognition has many important applications in science, law enforcement and entertainment. Popular methods to bridge the modality gap include reducing the distributional differences of representations of different modalities, learning indistinguishable representations or explicit modality transfer. The first two approaches suffer from the loss of discriminant information while removing the modality-specific variations. The third one heavily relies on the successful modality transfer, could face catastrophic performance drop when explicit modality transfers are not possible or difficult. To tackle this problem, we proposed a compact encoder-decoder neural module (cmUNet) to learn modality-agnostic representations while retaining identity-related information. This is achieved through cross-modality transformation and in-modality reconstruction, enhanced by an adversarial/perceptual loss which encourages indistinguishability of representations in the original sample space. For cross-modality matching, we propose MarrNet where cmUNet is connected to a standard feature extraction network which takes as inputs the modality-agnostic representations and outputs similarity scores for matching. We validated our method on five challenging tasks, namely Raman-infrared spectrum matching, cross-modality person re-identification and heterogeneous (photo-sketch, visible-near infrared and visible-thermal) face recognition, where MarrNet showed superior performance compared to state-of-the-art methods. Furthermore, it is observed that a cross-modality matching method could be biased to extract discriminant information from partial or even wrong regions, due to incompetence of dealing with modality gaps, which subsequently leads to poor generalization. We show that robustness to occlusions can be an indicator of whether a method can well bridge the modality gap.