I have a 6x6 matrix that equals the original matrix when multiplied by its transpose. What does this say about this matrix? What unique conditions does this matrix satisfy, since this property doesn't seem to hold in general?
If I understand the comments correctly, the matrices you're interested in have the following two properties:
Let me further assume that your matrices are real. Then these matrices are precisely the orthogonal projections onto some subspace (namely their image).