What specific sugars are found in apple cider, apple juice, and apple cider vinegar? My wife has a chronic, not-very-well-understood GI condition that includes hyper-sensitivity to certain foods.

We're looking to more information about the chemical composition of various foods for both healthcare and home-cooking purposes.

In particular, she has observed that:


She can consume both apples and apple cider (hard or regular) about as well as a typically healthy adult.
Apple juice causes her significant intestinal distress.
Even a tiny whiff of apple cider vinegar sets off her gastric symptoms immediately. (She can, however, safely consume other vinegars, such as white, red wine, balsamic, or rice wine vinegar.)


Are there chemical differences between apple cider, apple juice, and apple cider vinegar – such as transformations of the sugars due to fermentation of heating – that could explain this?
There certainly are chemical differences between apples and commercially purchased apple juice, but it's impossible to tell what they are, they are trade secrets. 

Apple juice bought in stores is not "apples squeezed then bottled", such a juice would become unappetizing before it was transported out of the factory. Juices are high tech products - "100% apple" just means that all ingredients were derived from apples, but the producer can have changed them in whatever ways it saw fit, before making a final product with the right consistency and aroma. 

If you are talking about freshly squeezed apple juice at home, the only chemical difference between that and whole apples is the oxidation of some antioxidants in the apple, mostly polyphenols. The sugars are not changed chemically.