Symmetric splitting of one-dimensional noises

A symmetric random walk $X$ whose jumps have diffuse law, looked at up to an independent geometric random time, splits at the minimum into two independent and identically distributed pieces. The same for the maximum. It is natural to ask, are there any other times adapted to $X$ exhibiting this "symmetric splitting"? It appears that the phenomenon is most conveniently couched in terms of (what may be called) the noise structure of $X$. At the level of generality of the latter, an equivalent set-theoretic condition for the symmetric splitting property is provided, leading to the observation that the answer to the elucidated question is to the affirmative. While we do not deal much with the obvious analog of the phenomenon in continuous time, the discrete findings do beg the question: does linear Brownian motion admit times of symmetric splitting other than the maxima and minima? This is left unresolved, but we do make some comments as to why it may be non-trivial/interesting.