Unitary equivalence of a matrix to its transpose

Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisingly, the naive assertion that a matrix is UET if and only if it is unitarily equivalent to a complex symmetric matrix (i.e., $T = T^t$) holds for matrices 7x7 and smaller, but fails for matrices 8x8 and larger.