  We use methods of the general theory of congruence and *congruence for
complex matrices--regularization and cosquares-to determine a unitary
congruence canonical form (respectively, a unitary *congruence canonical form)
for complex matrices A such that \bar{A}A (respectively, A^2) is normal.
  As special cases of our canonical forms, we obtain-in a coherent and
systematic way-known canonical forms for conjugate normal, congruence normal,
coninvolutory, involutory, projection, and unitary matrices. But we also obtain
canonical forms for matrices whose squares are Hermitian or normal, and other
cases that do not seem to have been investigated previously.
  We show that the classification problems under (a) unitary *congruence when
A^3 is normal, and (b) unitary congruence when A\bar{A}A is normal, are both
unitarily wild, so there is no reasonable hope that a simple solution to them
can be found.
