Mitsune and Motoko soul-swapped due to the Living Souls Exchange
Living Souls Exchange (生き血御霊写し, Iki Mitama Utsushi) is a secret skill of the Shinmei-ryu School of Swordsmanship. Normally, it is used during the process to remove an evil spirit from a person that is too weak to handle the operation. The person's spirit would be removed from the weak body and stored in a healthy one. Allowing the exorcist to destroy the evil spirits in the weak body without needing to worry about the strain that it may put on the person's weak spirit.
While this usually involves three people (the exorcist, the patient, and a volunteer) sometimes the exchange can occur simply between the exorcist and the patient; with the exorcist themselves taking control of the patient's body to exorcise it.
If an emergency occurs where the spirits cannot be returned to their bodies via the ritual, an emergency spell breaker exists where the exchanged souls must bath to purify themselves and then sleep together with the sword that conducted the ritual held between them. It should be noted, though, that this method only works in the first night of the exchange, afterward the souls will be trapped in their alternate bodies forever.