What are the best practices for reusing an animation controller?

Let's say I have several sprite sheets of characters that look different, but all have frames for the exact same types of movement. For example, "walking up/down/left/right" and "Idle up/down/left/right". Is there any way to easily reuse an animation controller that has already been configured to handle stuff like changing of directions and starting/stopping well or do I have to make a logically-redundant animator and swap out the sprites for each new spritesheet?