Women still haven't achieved equality and especially not pay equity in the far western countries.
Bullshit. Women have equal opportunity in every thing you can name and the "wage gap" is exclusively American rhetoric that every other Western nation has bought into when statistics repeatedly show that women, by their own choice, occupy careers that just pay less in general and it is illegal to pay based on sex.
The left does not want equal opportunity, though — they want equal outcome.
And if we were true right wingers, most of us wouldn't believe in evolution.
Your scientific beliefs have little sway on where you stand on the slidey scale of Left Wing and Right Wing. In fact, John Locke, "the father of liberalism" supported separation of Church and State. Secondly, liberalism is a right-wing ideology because classical liberalism supports rule of law and economic freedom under free-market capitalism. You'll find no shortage of libertarian "right wingers" who still support gay rights, scientific progress, etc. but are vehemently opposed to socialist policies.
On the opposite end, you can absolutely find hardcore religious types who support social programs and other state-funded endeavors and thus are "left wingers" but are authoritarian as f***, anti-science, anti-gay, etc. because Abrahamic religion is inherently authoritarian ideology.