There are several cards and powers that can change what skill is being used to pass a task: Incredible Ninja Leap uses Climbing for any skill, among others. ("I don't need to disguise myself to get past the Vatican tourists, I just leap over them from a nearby building!")
What does changing the skill you roll do to modifiers to the original skill roll? If your mission has a Combat test, and one of your opponents plays "Slipped!" on you to give -3 to your next Combat roll (i.e., this roll), but you change the skill to Climbing, does that -3 still apply to your roll? If there's a whole-mission penalty to Combat (such as "-2 to all Combat rolls on this mission"), does it still apply?