Self-inverse and exchangeable random variables

A random variable ZZ will be called self-inverse if it has the same distribution as its reciprocal Z−1Z−1. It is shown that if ZZ is defined as a ratio, X/YX/Y, of two rv’s XX and YY (with P[X=0]=P[Y=0]=0P[X=0]=P[Y=0]=0), then ZZ is self-inverse if and only if XX and YY are (or can be chosen to be) exchangeable. In general, however, there may not exist iid XX and YY in the ratio representation of ZZ.