I have heard many times that acid is added to a dish to "cut through" the richness, and conversely, that if one adds to much acid, this can be balanced out by the addition of a fat. Do acids and fats balance each other out like this and, if so, what is the mechanism or science behind it?
Fats have a bland taste and oily feel. Fatty foods can be made more tasteful by adding acids.
Acids stimulate taste, similarly as salt and, yes, they give the sensation of cutting through fat (Herbivoracious.com).
On the other hand, adding fats to acids (oil to vinegar in a salad) can decrease sharp acidic taste, but this happens only on a sensation level, not on a chemistry level, because fats do not absorb or chemically react with acids.
I'm not aware of any nutritional benefits of acid-fat mixtures. Acids also do not help significantly in fat digestion. Fats are relatively inert - they do not readily react with acids, alcohol...
Fat will inhibit the receptors in the taste buds; so it will change the experience of tasting. That also goes for sour/acidic tastes.