Let $X$ be a Binomial distribution with $n$ trials and success probability $p$ in $(0,1)$. It is clear that the quantiles of $X$ are an increasing function of $p$. Let $Y$ be a centered Binomial, i.e. $Y:=X-np$.
Can one also say something about whether $Y$'s quantiles are a monotone function of $p$?
Some thoughts: When $p=1/2$, the maximum ($p=100$) of $Y$ is $n/2$ while for $p\in \{0,1\}$, the maximum of Y is 0. So I don't expect uniform monotonicity. But I could imagine that the quantile function looks like the parabola $p(1-p)$: $0$ at $0$, then monotonically increasing until $1/2$ and then monotonically decreasing from $1/2$ to $1$.