This paper studies mutually coupled integrate-and-fire type chaotic oscillators. The coupling is realized by impulsive switchings and the system exhibits various synchronous and asynchronous phenomena. We give a basic classification of the chaos synchronization phenomena and their breakdown patterns. The stability of the synchronous states can be confirmed using the piecewise exact solutions, and the basic mechanism of the phenomena can be elucidated by a simple geometric consideration. The typical phenomena are confirmed in the laboratory.