I understand that such a question is rather left up to the DM. However that can come as a situation "DM against player" (which is not good but sometimes inevitable). So I'd like to know your interpretations and avoid being prejudiced.
Reverse Gravity (emphasis added):
This spell reverses gravity in a 50-foot-radius, 100-foot high cylinder centered on a point within range. All creatures and objects that aren’t somehow anchored to the ground in the area fall upward and reach the top of the area when you cast this spell. A creature can make a Dexterity saving throw to grab onto a fixed object it can reach, thus avoiding the fall. [...]
As I understood from the description, the spell changes the laws of physics in a certain area and causes objects and creatures in it to fall on the ceiling as if it were a floor.
Now, a creature with spider climb ability can climb difficult surfaces, including upside-down on ceilings, without needing to make an ability check.
I found two possible logical conclusions:
1st: While creatures, such as Giant Spiders, have spider climb and a climbing speed, they still have their normal movement speed, which is logical: while they are moving on the floor they are walking/standing. That makes them equal to all other normal walkers and the spell affects them the same as everyone.
2nd: The moment that the floor becomes ceiling, these creatures can instantly attach to it and become anchored to "the ground". After all, no actions needed to activate the spider climb as a creature walks from the wall to the ceiling. That's why spell doesn't affect them unless they are incapacitated/prone etc.