  Given a permutation pattern p and an equivalence relation on permutations, we
study the corresponding equivalence classes all of whose members avoid p. Four
relations are studied: Conjugacy, order isomorphism, Knuth-equivalence and
toric equivalence. Each of these produces a known class of permutations or a
known counting sequence. For example, involutions correspond to conjugacy, and
permutations whose insertion tableau is hook-shaped with 2 in the first row
correspond to Knuth-equivalence. These permutations are equinumerous with
certain congruence classes of graph endomorphisms. In the case of toric
equivalence we find a class of permutations that are counted by the Euler
totient function, with a subclass counted by the number-of-divisors function.
We also provide a new symmetry for bivincular patterns that produces some new
non-trivial Wilf-equivalences
