In the ordered set $R_0=\{1,2,\dots , p-1\}$, where $p$ is prime, the additive inverse of $a$ with respect to the modulus $p$ is simply $(p-a)$, and the pair $a,(p-a)$ are positionally separated by $|p-2a|$ steps. For example, for $a=2,p=7$, one counts $7-2\cdot 2=3$ times starting from $2$ to get to its additive inverse $5$, viz: $3,4,5$. Moreover, to get from $5$ to $2$, one counts $6,1,2$, three times (wrapping around), taking into account that $0\equiv p$ is not in the set.
I am interested in what happens with regard to the positional separation of additive inverses when the members of the set are reordered in a systematic way. Specifically, for a particular $1<a<(p-1)$ and $0\le n \le (p-1)$ and some positive integer constant $d$ such that $p\not \mid d$, we create the ordered set $R_1=\{a,a+d,\dots,a+nd, \dots,a+(p-1)d\}$.
To put a context on all of this, in a problem I am working on, what statements can be reliably asserted depends on how small the difference between additive inverses are in certain circumstances. The largest difference between additive inverses in $R_0$ is $p-2$. The element in $R_1$ which is congruent to the additive inverse of $a$ may be very much larger, i.e. $a+n_kd-a=n_kd$.
An arithmetic progression having $p$ elements will have one element which is divisible by $p$, so we exclude the sole element $a+n_xd \equiv 0 \bmod p$. Since $n$ can take on $p$ values, but one of the elements is excluded, $R_1$ has $p-1$ members.
With respect to modulus $p$, $R_1$ has each of the members of $R_0$. If any two members were the same, then their difference would be $\equiv 0 \bmod p$. $$a+n_1d-a+n_2d=(n_1-n_2)d \equiv 0 \bmod p \Rightarrow p\mid (n_1-n_2) \Rightarrow n_1=n_2$$ Thus, if their difference is $\equiv 0 \bmod p$, they are the same member, not different members. If no two members are the same, and there are $p-1$ members, then they are the same as the members $\bmod p$ that are found in $R_0$ except for order.
I have played around with several small values of $a,d,p$, looking at $a$ and its additive inverse $p-a$, but I can observe no readily formulable relationship describing the positional separation of those numbers in the reordered set(s). Nor have I been able to derive a mathematical statement that describes such.
My question is: Is there a formulable relationship regarding the positional separation of $a$ and its additive inverse $p-a$ in the ordered set $R_1$ described above?