Another approach to try might be to split the matrices into there individual elements:
The inverse of a 2x2 matrix [O] is:
[O]^{1} = 1/(o00 o11 * o01 o10) 

so expanding out [O]^{T} = [O]^{1} gives:
o00 = o11/(o00 o11 * o01 o10)
o10 = o01/(o00 o11 * o01 o10)
o01 = o10/(o00 o11 * o01 o10)
o11 = o00/(o00 o11 * o01 o10)
If we constrain the determinant to be 1 then,
(o00 o11 * o01 o10) = 1
o00 = o11
o10 = o01
we are looking for C where [O] = [C]*[M]