  Given two orthogonal projections P and Q, we are interested in all unitary
operators U such that UP=QU and UQ=PU. Such unitaries U have previously been
constructed by Wang, Du, and Dou and also by one of the authors. One purpose of
this note is to compare these constructions. Very recently, Dou, Shi, Cui, and
Du described all unitaries U with the required property. Their proof is via the
two projections theorem by Halmos. We here give a proof based on the
supersymmetric approach by Avron, Seiler, and one of the authors.
