  This paper concerns the enumeration of rotation-type and congruence-type
convex polyominoes on the square lattice. These can be defined as orbits of the
groups C4, of rotations, and D4, of symmetries of the square acting on
(translation- type) polyominoes. In virtue of Burnside's Lemma, it is
sufficient to enumerate the various symmetry classes (fixed points) of
polyominoes defined by the elements of C4 and D4. Using the
Temperley--Bousquet-Melou methodology, we solve this problem and provide
explicit or recursive formulas for their generating functions according to
width, height and area. We also enumerate the class of asymmetric convex
polyominoes, using Moebius inversion, and prove that their number is
asymptotically equivalent to the number of convex polyominoes, a fact which is
empirically evident.
