It's not that simple.
The right, for instance, very much does NOT support gender neutrality, and in fact, that's the exact reason that they call for gender-neutral physical standards for the military. The actual goal is to set standards that are more easily attainable by biological males in order to generally keep women out of the armed forces.
And in the same sense it can be argued that the left actually is gender neutral regarding the military, which is why they want physical standards that are relative to the norms for biological males and biological females respectively rather than one standard that favors biological males. Their gosl is gender neutrality in overall composition of the military, which, due to innate biological differences, requires differing standards along the way.
So both are "neutral" situationally, but neither is broadly. And that's the case for all ideologies - they might argue for some sorr of gender neutrality when ir serves their broader goal of forcing society into whatever form they prefer, but as soon as actual neutrality would conflict with their goals, they'll just as easily forego it.