Confidence Intervals guaranteed to be centered at 0? With most pivotal quantities that I'm thinking of, after we rearrange stuff to get an interval, the interval's "location" depends on a random point estimate. The example below has an interval with midpoint equal to the sample mean.
Is there a way to guarantee that we end up with an interval centered at $0$? By "centered at $0$" I mean an interval of the form $(-c,c)$ for some fixed $c \in \mathbb{R}^+$.
For example, with a normal random sample, we could start with the pivotal quantity $\sqrt{n}(\bar{x} - \mu)/s$, and rearrange the right hand side of
$$
1 - \alpha = \mathbb{P}\left(-t_{\alpha/2,n-1} \le \sqrt{n}(\bar{x} - \mu)/s \le t_{\alpha/2,n-1} \right)
$$
into
$$
\mathbb{P}\left(\bar{x}-\frac{s}{\sqrt{n}}t_{\alpha/2,n-1} \le \mu \le \bar{x}+\frac{s}{\sqrt{n}}t_{\alpha/2,n-1}\right).
$$
Is there a way we could scoot it over $\bar{x}$ and then fix the (random) width so that we end up with $95$% coverage, still?
I suspect it isn't possible. If I'm not changing the pivotal quantity, all I can do is pick the quantiles. This example is an "equal-tailed" interval. But how can I deterministically pick quantiles, by picking two positive numbers that sum to $\alpha$, based on a random data set I haven't obtained yet?

I prefer visualizing frequentist inference using a confidence curve.

Without loss of generality we could consider a hypothesis $H_0: \theta=\theta_0$ (not necessarily zero).  Above is a confidence curve for inference on a Bernoulli proportion $p$ from inverting the binomial CDF, $F_Y(y,n,p)$, based on $6$ events with a sample size of $n=10$.  Considering the hypothesis $H_0:p=0.5$, for this experimental result the upper-tailed p-value is $0.38$.  We are therefore $38\%$ confident the unknown fixed true $p$ is less than or equal to $0.5$ and $100(1-0.38)\%=62\%$ confident the unknown fixed true $p$ is greater than or equal to $0.5$.  These confidence levels are nothing more than a restatement of the one-sided p-value testing $H_0: p= 0.5$.  The vertical reference lines identify the two-sided equal-tailed $75\%$ confidence interval.  This is the set of hypotheses that are most plausible given the observed data $-$ those hypotheses that when tested are not significant at the two-sided equal-tailed $25\%$ level.  For each hypothesis in this interval the observed result is within a $75\%$ margin of error. Here is a related post.
For the given experimental result, by considering a margin of error that does not have equal tails you can construct a $95\%$ confidence interval that is centered at $H_0:p=0.5$.
For the same experimental result here is another confidence curve centered at $H_0: p=0.5$.  The maximum likelihood estimate is still $\hat{p}=0.6$ but I have defined the peak of the curve at $p=0.5$.  The $95\%$ confidence interval centered at $0.5$ is $(0.15, 0.85)$  The upper-tailed p-value testing $H_0: p=0.15$ is $0.001$ and the lower-tailed p-value testing $H_0: p=0.85$ is $0.049$.  We are therefore $95\%$ confident the unknown fixed true $p$ is within $(0.15, 0.85)$.

