There are actually many examples of things working different for the mirror image, e.g. many medical/chemical compounds work differently based on chirality.
The topic of chirality is fascinating, one of the big puzzles in nature is for example why it so strongly prefers right-handedness.
It "works different" because you consider only change in one part of system. You change compound, but rest of system stays the same. If you mirror entire system, it works almost the same way (differences due to parity violation by weak force are millions time smaller than required to even detect differences).