I wanted to add to the above some general explanation that:

Antigliadin antibodies show an abnormal immune system response to dietary gluten.

Anti-tTG (anti-TG2) antibodies show an abnormal immune system response targeting the intestinal villi. These are "autoantibodies" that directly attack the gut. This is what makes celiac disease an autoimmune disease.

Anti-TG6 are the new autoantibodies discovered associated with gluten ataxia and other neurological manifestations of gluten sensitivity, but they are not yet available commercially.

IgA class antibodies are more associated with gut manifestations of gluten sensitivity. IgG class antibodies are more associated with non-gut manifestations. This seems to hold true whether looking at antigliadin or anti-tTG antibodies.

So, the problem is the antibodies most relevant for celiac disease are least relevant when looking at a neurological manifestition of gluten sensitivity, and vice versa.

Most specific for celiac disease> anti-tTG IgA

Most specific for gluten related neurological disease> antigliadin IgG

This causes lots of problems when doctors are too focused on celiac disease and not the broader reaches of gluten sensitivity.