  We revisit the complexity of deciding, given a {\it bimatrix game,} whether
it has a {\it Nash equilibrium} with certain natural properties; such decision
problems were early known to be ${\mathcal{NP}}$-hard~\cite{GZ89}. We show that
${\mathcal{NP}}$-hardness still holds under two significant restrictions in
simultaneity: the game is {\it win-lose} (that is, all {\it utilities} are $0$
or $1$) and {\it symmetric}. To address the former restriction, we design
win-lose {\it gadgets} and a win-lose reduction; to accomodate the latter
restriction, we employ and analyze the classical {\it
${\mathsf{GHR}}$-symmetrization}~\cite{GHR63} in the win-lose setting. Thus,
{\it symmetric win-lose bimatrix games} are as complex as general bimatrix
games with respect to such decision problems. As a byproduct of our techniques,
we derive hardness results for search, counting and parity problems about Nash
equilibria in symmetric win-lose bimatrix games.
