The phenomenon where a speaker mixes two or more languages within the same conversation is called code-switching (CS). Handling CS is challenging for automatic speech recognition (ASR) and text-to-speech (TTS) because it requires coping with multilingual input. Although CS text or speech may be found in social media, the datasets of CS speech and corresponding CS transcriptions are hard to obtain even though they are required for supervised training. This work adopts a deep learning-based machine speech chain to train CS ASR and CS TTS with each other with semisupervised learning. After supervised learning with monolingual data, the machine speech chain is then carried out with unsupervised learning of either the CS text or speech. The results show that the machine speech chain trains ASR and TTS together and improves performance without requiring the pair of CS speech and corresponding CS text. We also integrate language embedding and language identification into the CS machine speech chain in order to handle CS better by giving language information. We demonstrate that our proposed approach can improve the performance on both a single CS language pair and multiple CS language pairs, including the unknown CS excluded from training data.