  A novel method to obtain parametrizations of complex inverse orthogonal
matrices is provided. These matrices are natural generalizations of complex
Hadamard matrices which depend on non zero complex parameters. The method we
use is via doubling the size of inverse complex conference matrices. When the
free parameters take values on the unit circle the inverse orthogonal matrices
transform into complex Hadamard matrices, and in this way we find new
parametrizations of Hadamard matrices for dimensions n= 8,10,12.
