We study the problem of multilingual masked language modeling, i.e. the training of a single model on concatenated text from multiple languages, and present a detailed study of several factors that influence why these models are so effective for cross-lingual transfer. We show, contrary to what was previously hypothesized, that transfer is possible even when there is no shared vocabulary across the monolingual corpora and also when the text comes from very different domains. The only requirement is that there are some shared parameters in the top layers of the multi-lingual encoder. To better understand this result, we also show that representations from independently trained models in different languages can be aligned post-hoc quite effectively, strongly suggesting that, much like for non-contextual word embeddings, there are universal latent symmetries in the learned embedding spaces. For multilingual masked language modeling, these symmetries seem to be automatically discovered and aligned during the joint training process.