Given three real numbers $a, b$ and $t$ with $t$ positive, let $\beta$ be a one-dimensional Brownian bridge of length $t$ from $a$ to $b$. In this talk, based on a conditional identity in law between Brownian bridges stemming from Pitman's theorem, we show that the process given by \begin{align} \beta_{t-s}+\biggl| b-a+ \min _{0\le u\le t-s}\beta_{u}-\min _{t-s\le u\le t}\beta_{u} \biggr| -\biggl| \min _{0\le u\le t-s}\beta_{u}-\min _{t-s\le u\le t}\beta_{u} \biggr| \end{align} for $0 \le s \le t$, has the same law as $\beta$. The path transformation that describes the above process is proven to be an involution, commute with time reversal, and preserve a Pitman-type transformation in conjunction with time reversal. Since it does not change the minimum value in particular, the transformation also preserves the law of a three-dimensional Bessel bridge of length $t$. As an application, some distributional invariances of three-dimensional Bessel processes are derived. This talk is based on arXiv:2503.06813.