Context. Coronal bright points (CBPs) are small-scale and long-duration brightenings in the lower solar corona. They are often explained in terms of magnetic reconnection. Aims. We aim to study the substructures of a CBP and clarify the relationship among the brightenings of different patches inside the CBP. Methods. The event was observed by the X-ray Telescope (XRT) aboard the Hinode spacecraft on 2009 August 22−23. Results. The CBP showed repeated brightenings (or CBP flashes). During each of the two successive CBP flashes, that is, weak and strong flashes that were separated by ∼2 hr, the XRT images revealed that the CBP was composed of two chambers, patches A and B. During the weak flash, patch A brightened first, and patch B brightened ∼2 min later. During the transition, the right leg of a large-scale coronal loop drifted from the right side of the CBP to the left side. During the strong flash, patch B brightened first, and patch A brightened ∼2 min later. During the transition, the right leg of the large-scale coronal loop drifted from the left side of the CBP to the right side. In each flash, the rapid change of the connectivity of the large-scale coronal loop is strongly suggestive of the interchange reconnection. Conclusions. For the first time we found reciprocatory reconnection in the CBP, which means that reconnected loops in the outflow region of the first reconnection process serve as the inflow of the second reconnection process.