Suppose $p$ is prime. Given an integer $x$ with $0 < x < p$, say $x$ is reversible from $i$ to $y$ precisely when $p - 1 - x = (p \times i - 1) \pmod y$.
Some examples with $p=7$:
$1$ is not reversible from $-9$
$6$ is reversible from $-5$ to any of the $y$ in $\{1,2,3,4,6\}$
$5$ is reversible from $-7$ to the unique $y$ in $\{3\}$
Now say $x$ is "special" exactly when there is an arbitrarily long sequence $y_0, y_1, y_2,...$ where $x$ is reversible from $0$ to $y_0$ and each $y_i$ is reversible from $-(i+1)$ to $y_{i+1}$.
My question: given a prime $p$, is there more than one such special $x$?