Ring structure of the units of order $2$ in a monogenic order?

Ring structure of the units of order $2$ in a monogenic order?

A wise man once told me the following:
Let $f\in\mathbb{Z}[X]$ be monic and let $R=\mathbb{Z}[X]/(f)$, so that
$(R,+,\times)$ is a ring. Let $$U=\{u\in R\mid u^2=1\}.$$ Then
$(U,\times,\star)$ is a ring, with multiplication satisfying $$u\star
v=w\star x\quad\Leftrightarrow\quad\ (1-u)(1-v)=(1-w)(1-x).$$
I haven't the faintest clue how $\star$ could be defined. Any ideas?