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Recently, user-oriented auto-encoders (UAEs) have been widely used in recommender systems to learn semantic representations of users based on their historical ratings. However, since latent item variables are not modeled in UAE, it is difficult to utilize the widely available item content information when ratings are sparse. In addition, whenever new items arrive, we need to wait for collecting rating data for these items and retrain the UAE from scratch, which is inefficient in practice. Aiming to address the above two problems simultaneously, we propose a mutually-regularized dual collaborative variational auto-encoder (MD-CVAE) for recommendation. First, by replacing randomly initialized last layer weights of the vanilla UAE with stacked latent item embeddings, MD-CVAE integrates two heterogeneous information sources, i.e., item content and user ratings, into the same principled variational framework where the weights of UAE are regularized by item content such that convergence to a non-optima due to data sparsity can be avoided. In addition, the regularization is mutual in that user ratings can also help the dual item content module learn more recommendation-oriented item content embeddings. Finally, we propose a symmetric inference strategy for MD-CVAE where the first layer weights of the UAE encoder are tied to the latent item embeddings of the UAE decoder. Through this strategy, no retraining is required to recommend newly introduced items. Empirical studies show the effectiveness of MD-CVAE in both normal and cold-start scenarios. Codes are available at https://github.com/yaochenzhu/MD-CVAE.