Abstract We adapt C. Freiling's axioms of symmetry (J. Symbolic Logic 51 (1986) 190–200) to models of set theory with classes by identifying small classes with sets getting thus a sequence of principles A n , for n⩾2, of increasing strength. Several equivalents of A 2 are given. A 2 is incompatible both with the foundation axiom and the antifoundation axioms AFA ∼ considered in Aczel (Non Well Founded Sets, CSLI Lecture Notes, vol. 14, Stanford University, 1988). A hierarchy of symmetry degrees of preorderings (and of classes carrying such preorderings) is introduced and compared with A n . Models are presented in which this hierarchy is strict. The main result of the paper is that (modulo some choice principles) a class X satisfies ¬ A n iff it has symmetry degree n−2.