  Let X be a pseudocyclic association scheme in which all the nontrivial
relations are strongly regular graphs with the same eigenvalues. We prove that
the principal part of the first eigenmatrix of X is a linear combination of an
incidence matrix of a symmetric design and the all-ones matrix. Amorphous
pseudocyclic association schemes are examples of such association schemes whose
associated symmetric design is trivial. We present several non-amorphous
examples, which are either cyclotomic association schemes, or their fusion
schemes. Special properties of symmetric designs guarantee the existence of
further fusions, and the two known non-amorphous association schemes of class 4
discovered by van Dam and by the authors, are recovered in this way. We also
give another pseudocyclic non-amorphous association scheme of class 7 on
GF(2^{21}), and a new pseudocyclic amorphous association scheme of class 5 on
GF(2^{12}).
