Term for perfect symmetric asymmetry
A game where the payoff function $F_1,F_2$ are such that for any two strategy profiles $s,s'$
$$
F_1(s) &gt; F_1(s') \Leftrightarrow F_2(s) &lt; F_2(s')
$$
is called an antagonistic game. Sounds like what you want is a symmetric antagonistic game?