An idempotent with respect to some modulus $n$ is a number that remains unchanged when it is raised to any positive integer power: $a^k\equiv a \bmod n \iff a\ \text{is an idempotent}$. As a concrete example, $6^k\equiv 6 \bmod 10$.
It is readily appreciated that the additive inverse of an idempotent will exhibit a related repeating behavior: $(-a)^{2m}\equiv a \bmod n$ and $(-a)^{2m+1}\equiv -a \bmod n$. For example, $4$ is the additive inverse of $6 \bmod 10$ and $4^k \bmod 10$ oscillates between $6$ and $4$. I note for the sake of thoroughness there are cases where and idempotent and its additive inverse are the same, such as $7^k\equiv (-7)^k \equiv 7 \equiv (-7) \bmod 14$.
My (very soft) question: Is there a term to refer to the additive inverse of an idempotent (other than the phrase I just used, which discloses nothing of the behavior I refer to), or terminology to refer to the oscillatory behavior it displays when exponentiated? I looked around on various sites for an hour or so and came up with nothing on point.