Subdiagrams equal in number to their duals

In the middle 1930's, the early days of combinatorial lattice theory, it had been conjectured thatin any finite modular lattice the number of join-irreducible elements equals the number of meetirreducible elements. The conjecture was settled in 1954 by R. P. Dilworth in a remarkable combinatorial generalization. Quite recently we have been led to this question. Which ordered setsS satisfy the property that,for any modular lattice M, the number of subdiagrams of M isomorphic to S equals the number of subdiagrams of M dually isomorphic to S?