Each Rating has Anti-Rating
Letter "E" may mean "Economic" or "Enhanced" or something other proper word. While working with E-Rating I saw that I have two absolutely equivalent ways to calculate rating.
Usually we calculate rating as measure of strength. But in the same way we could calculate "anti-rating" as measure of weakness. For this we have to replace "win" with "loss" and "loss" with "win". As a result of this operation we will obtain another rank-list. Ranking by anti-rating will not exactly be congruent with the Ranking by rating.
I think that "rating mirror" could be calculated for all ranking systems. As well as both rank-list are equivalent the problem is how to join rank-lists by "rating" and "anti-rating".
It depends on rating system. In general case we have to calculate two probabilities to win using both rating and anti-rating and then calculate mean value.
This way gives us probabilities but it does not give unit rank-list. Another way is to create artificial value such as sqr(Ra/ARa). Kenneth Massey proposed it when we discuss this problem.
Let's forget mathematical point of view on ranking problems and try to solve it on other way.
Let's think that results of matches are goods. If teams wins it means that this team buys a game. If teams lose it means that it sells a game. Naturally that price for win and loss depends on strength of team.