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Yes, as a sweeping generalization, horses can climbstairs. There'll be exceptions where the width and/or steepnessprevent them from doing so, and any designs that make us balk atthe idea of climbing them are a definite no-no, but horsesare incredibly adaptable. Going up is much easier thangoing down.
Furthermore, can pigs go down stairs?
If pigs are climbing stairs, they may getdaring and start to leap off the last step or jump on andoff furniture. This could not only cause a pinched nerve, but alsoa broken leg or back. Pigs do not climb or jump inthe wild, so by nature, they aren't meant to climbstairs.
Herein, can you be backwards on stairs?
You can't stand backwards on astair. You can walk backwards on stairs,though. If you're walking up stairs, then stop andmove backwards back down, are you ever standingbackwards?
You can't expect a horse to be able toclimb a ladder. Perhaps a harness and rope systemcould be devised to lower the animal down, but in thisinstance the climb skill for the creature can't beused, or it needs to have a large negative modifier.