In this paper, we present a multimodal and dynamical VAE (MDVAE) applied to unsupervised audio-visual speech representation learning. The latent space is structured to dissociate the latent dynamical factors that are shared between the modalities from those that are specific to each modality. A static latent variable is also introduced to encode the information that is constant over time within an audiovisual speech sequence. The model is trained in an unsupervised manner on an audiovisual emotional speech dataset, in two stages. In the first stage, a vector quantized VAE (VQ-VAE) is learned independently for each modality, without temporal modeling. The second stage consists in learning the MDVAE model on the intermediate representation of the VQ-VAEs before quantization. The disentanglement between static versus dynamical and modality-specific versus modality-common information occurs during this second training stage. Extensive experiments are conducted to investigate how audiovisual speech latent factors are encoded in the latent space of MDVAE. These experiments include manipulating audiovisual speech, audiovisual facial image denoising, and audiovisual speech emotion recognition. The results show that MDVAE effectively combines the audio and visual information in its latent space. They also show that the learned static representation of audiovisual speech can be used for emotion recognition with few labeled data, and with better accuracy compared with unimodal baselines and a state-of-the-art supervised model based on an audiovisual transformer architecture.