  For real matrices selfadjoint in an indefinite inner product there are two
special canonical Jordan forms, that is (i) flipped orthogonal (FO) and (ii)
$\gamma$-conjugate symmetric (CS). These are the classical Jordan forms with
certain additional properties induced by the fact that they are
$H$-selfadjoint. In this paper we prove that for any real $H$-selfadjoint
matrix there is a $\gamma$-FOCS Jordan form that is simultaneously flipped
orthogonal and $\gamma$-conjugate symmetric.
