Drag the sample mean and watch the rejection region and the confidence interval flip their verdicts at the same instant. They have to: both are built from the same standard error and the same critical value.
The cutoffs sit one margin of error either side of μ₀, and the interval reaches one margin of error either side of x̄. That is the same distance measured from opposite ends — so “x̄ landed in the red” and “the interval missed μ₀” can never disagree.