I would say a cenetaur could attempt to climb a rope or a cliff, but I would factor in that in that phase, his front legs are poorly manuvered (-2 for that), and rear end become dead weight, if ground cannot support his rear half in the way he is climbing, then that should be added to his carying capacity, (which also should be cut down to a bi-peds carying capacity while unable to keep his feet on the ground. Going with the 1,000 LB weight of a horse, lets subtract 200 for the upper half. Considering the description for climb says that if you catch someone else, that exceeds your heavy load capacity, you automatically fall, I would say a centaur, must have a minimum of 26 str to even attempt to climb anything they can't have a solid footing for all 4 legs (or higher if he's wearing armor or carrying anything else that would knock him out of the 694–1,040 lbs. bracket), and of course all penelties for a heavy load, should apply.
I admit it might seem funny to imagine a centaur climbing a cliff but not any more so than a four foot high halfling doing the same thing.
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the hoof is no less able to find a purchase than a booted foot.
"Not any more so"? "No less able"?
Yeah, I'm afraid you've lost me.
Factor in a penalty on climb checks that seem out of place. Climbing the wall with a rope like batman? Penalty or no penalty? Trying to climb the same wall without the aid of a rope, impose a difficulty to it. They "can" climb, it is just going to be hard as heck.
Something akin to an armor check penalty, I suppose.