As AI-generated content (AIGC) thrives, deepfakes have expanded from single-modality falsification to cross-modal fake content creation, where either audio or visual components can be manipulated. While using two unimodal detectors can detect audio-visual deepfakes, cross-modal forgery clues could be overlooked. Existing multimodal deepfake detection methods typically establish correspondence between the audio and visual modalities for binary real/fake classification, and require the co-occurrence of both modalities. However, in real-world multi-modal applications, missing modality scenarios may occur where either modality is unavailable. In such cases, audio-visual detection methods are less practical than two independent unimodal methods. Consequently, the detector can not always obtain the number or type of manipulated modalities beforehand, necessitating a fake-modality-agnostic audio-visual detector. In this work, we introduce a comprehensive framework that is agnostic to fake modalities, which facilitates the identification of multimodal deepfakes and handles situations with missing modalities, regardless of the manipulations embedded in audio, video, or even cross-modal forms. To enhance the modeling of cross-modal forgery clues, we employ audio-visual speech recognition (AVSR) as a preliminary task. This efficiently extracts speech correlations across modalities, a feature challenging for deepfakes to replicate. Additionally, we propose a dual-label detection approach that follows the structure of AVSR to support the independent detection of each modality. Extensive experiments on three audio-visual datasets show that our scheme outperforms state-of-the-art detection methods with promising performance on modality-agnostic audio/video deepfakes.