Summary: The \((2^w)!\) reversible logic gates of width \(w\), i.e. reversible logic gates with \(w\) inputs and \(w\) outputs, together with the action of cascading, form a group \(R\). We define a subgroup \(K\), consisting of the SWITCHED gates. There are \([(2^{w-1})!]^2\) such gates. They partition the group \(R\) into \(2^{w-1} + 1\) double cosets. This allows us to decompose an arbitrary reversible gate into the cascade of three gates: a SWITCHED gate, an upside-down simple control gate and a second SWITCHED gate. We present an algorithm to perform this factorization, and thus provide a method of implementing an arbitrary reversible gate into hardware. The algorithm can be used to automate the implementation of a reversible function in future (c-MOS) technologies, realizing low-cost computing.