There will often be some friction at various parts of the tuning linkage, as well as at the nut (where the string passes over). At some points, including the nut, things may bind slightly. Think about what would happens if the string is binding where it passes over the nut, both in the "tightening" and "loosening" cases.
If the string binds where it passes over the nut while the player is loosening it, then the portion of the string between the peg and the nut may be at lower tension than the main sounding portion of the string. Each time the string is plucked, its tension will be momentarily increased. Consequently, even if the tension difference at the nut wasn't sufficient to cause slippage while the string was resting, the act of playing the string might cause it to slip at some inopportune time.
If, by contrast, the last action was to tighten the string, then the portion of the string connecting the peg will if anything be tighter than the sounding portion. If it's not "enough tighter" to overcome friction when the string is resting, it will be even less inclined to move when the main string is drawn more tightly (reducing the tension difference).
Although it's easiest to visualize the above behavior happening at the nut, it can also happen at other parts of the mechanical linkage. For example, the peg might bind slightly such that it would stay put (at least for awhile) even if the worm cased to exist. Ideally, the tuning gear would always ride against the worm, but if the peg binds, it might not turn to loosen the string unless or until the worm pushes it. If one turns the worm enough to turn the peg 0.4 degrees, but the peg binds after turning only 0.3 degrees, then one would initially hear the string's pitch as though the worm had been tuned 0.3 degrees, but if the peg slips (as it likely would at some point) the tuning would change by an extra 0.1 degree.
As a principle, what's important is not whether the pitch of something is being raised or lowered, but rather whether one is pushing something in the direction opposite where it wants to be. For most instruments, that would represent raising the pitch, but some electronic instruments which are tuned by adjusting inductors work in the opposite direction; I wouldn't be surprised if some mechanical acoustic instruments do so as well.