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No — rounding never touches your actual start or end time, only the tracked duration that gets rounded to your chosen interval.
You get to choose where that rounding is applied:
There’s one separate setting worth knowing about, though: turning off “Exact tracking” in favor of “Without seconds” tracks start and end times to the minute rather than the second. That’s not a form of rounding — it’s a precision setting for how exactly a start or end time is captured in the first place.