According to some of the logic on here, there are no rules in the MM that says they can't climb. In fact, they get +6 to their athletics.
And the MM states very clearly, that a climb speed simply means they don't have the penalty associated with climbing. That is, unless we are assumed to believe that if it is left off, it can't climb at all.
But that wouldn't make sense, because goats wouldn't be able to climb. You know, because they don't have a climb speed. And deer and elk wouldn't be able swim either because they don't have a swim speed. But I have seen them swim as well as bears and elephants. But none of them have a swim speed. And by that logic, a constrictor snake wouldn't be able to climb. And frogs can't climb! Again, no climb speed listed. Man, the MM is crazy!
Or they intended people to use common sense, and were like, no DM is going to have a centaur climb a castle wall. But we showed them!