Choice and Regularity: Common Consequences in Logic

It is well-known that Choice and Regularity are independent of each other but have important common consequences of logical character (reflection principles, representations of classes by sets, etc.). We explain this phenomenon by isolating their "intersection", a principle (called here Best-Foundedness) which is consistent with the negations of both axioms but implies all these consequences. Then we study relationships between these consequences (and near principles) in detail. Finally, we consider some arguments related to truth of various principles in set theory, especially arguments concerning the interpretability strength.