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?
 A: The obvious thing to do is to define $u \star v = (1-u)(1-v)$.  This satisfies the last condition, but unfortunately, it doesn't work: This $\star$ is commutative and associative, but it does not distribute over the original $\times$ in $R$.  However, it does suggest that we might be able to modify this definition slightly.
Starting with the last condition, if you take $u=w=1$, then the condition implies that $1 \star v = 1 \star x$ for all $v,x$, so clearly the original 1 in the ring must be the additive identity in the new ring.  Further if you consider the case $f=x$, then $U = \{1,-1\}$ which suggests that the original -1 must be the multiplicative identity in the new ring.  In other words, we should have $(-1) \star v = v$ for all $v$, which suggests the definition $$u \star v = \frac{2-(1-u)(1-v)}{2}.$$
Since $x \mapsto (2-x)/2$ is a bijection on $R$, this still satisfies the last condition, and you can check that this does distribute over $\times$, and is commutative and associative.
