Mostly this is an excuse to link to these great videos of free running and parkour, unusual because they show much of the full sequence rather than mash-ups. But to go all scholarly on you, Cognitive Daily had a recent piece on learning to walk and children’s sense of balance. Leaning with backpack weights was a learned process, not an intuitive one, even with toddlers who knew how to walk.
These videos also give me the chance to plug Greg’s early piece on our sense of balance. Rather than an innate module gifted to us by evolution, “The evidence seems very clear that the sense of balance (again, with all the caveats of calling it ‘a’ single ‘sense’) can be trained to wide range of different challenges and to operate more efficiently or from different sets of information depending upon the task constraints. The variability of equilibrium was driven home to me in my research on capoeira, an Afro-Brazilian martial art and dance.”
Similarly with “l’art de displacement” through mixing balance, jumping, climbing and running. Wow!
Just like elite runners, I bet they stay focused on the task on hand, and not on the pain of a misstep or the fear over a missed jump—dissociation from risk and worry through expert technique. And this focused and skilled activity also relies on significant sensory integration of balance, vision, and touch. In turn, sensory integration, plenty of training and experience, and focus on the task help make free running predictable, understandable and controllable, and thus integrated into the person’s everyday interactive design.
Anyways, here’s a couple popular YouTube videos in the mash-up music video style:
Neuroanthropology 
I share your hesitation to call this a ‘sense’, but it should be noted that this also holds for much of the SAE folk model of the five senses (neurophysiologically, there are more like 20+ distinct senses). So much for the neuro part of the story — on the anthropology side, I’m sure you’re aware of the fascinating work done under the broad heading of ‘anthropology of the senses’. On the ‘sense of balance’ specifically, there are the studies by Geurts (2002a,b) of the culturally constructed sensorium in an African community, where this ‘sense of balance’ is much more elaborated than in some other cultural settings (think headloading).
* Geurts, Kathryn Linn. 2002a. On Rocks, Walks, and Talks In West Africa: Cultural Categories and an Anthropology of the Senses. Ethos 30, no. 3: 178-198. doi:doi:10.1525/eth.2002.30.3.178.
* Geurts, Kathryn Linn. 2002b. Culture and the Senses: Bodily ways of knowing in an African community. Berkeley: University of California Press.