To underscore my comments posted under another answer, here is a start of an enumeration which would be pretty messy when completed.
I will cover the case of 4-fold rotational symmetry. I consider the lower left 3x3 squares which I label with coordinate pairs (1,1) through (3,3). (3,3) is a central square which by symetry I assume to be part of any shape I count. Since I am interested in connected shapes, one of (3,2) or (2,3) (squares below and/or to the left) must belong to the shape as well. Suppose both are: then there are 64 cases to consider, where each of the remaining 6 squares of the lower left 3x3 square is part of the included shape or not.
Suppose square (2,2) is not part of the shape. Then one of its three equivalent (under rotation) squares must be. Some thought will show that it is not the (5,5) square. Say it is the (5,2) square, then at least two more squares will be needed to connect it to the
squares in the lower left corner. so at most 3 more squares are allowed in our shape from
the lower left corner.
I could go on using words; it would be easier to list the allowed shapes that I discovered by trial and error, and submit that as a list. It is possible that an alternate categorization (counting those where (1,1) and/or (2,2) are part of the shape, for example) might lead to a more concise counting, but most people who would need the count would probably spend the time trying to derive it for themselves rather than read your derivation. Again, I do not see a solution to this problem that is quick and easy with few cases to consider.
Gerhard "Ask Me About System Design" Paseman, 2011.05.21