Hi

When I move the fader to the left and press Correct Tuning, the sound is an octave lower.
When I move the fader up, it doesn't happen.

The pitch correction algorithm "listens" to the sound using the current tuning (it simply plays the C3 note) and then figures out a tuning adjustment. If the adjusted tuning "fits" on the Tuning slider, it will use it. Otherwise, it will wrap and update a secret broader pitch offset value inside the patch.

I guess when you tune it first with the high pitch, the algorithm wants to tune it a little lower (thus, the Tuning slider remains almost identical). When you tuned it with the low pitch, it also wanted to go a little lower, but it couldn't do so without resetting the tuning slider.