Both the left and the right claim to support equality, but their definitions differ.
The right tends to advocate for equality of opportunity, while fully acknowledging the natural differences between each person. The left tends to advocate for equality of outcome. This second definition is what is referred to as equality in this dichotomy, being the antithesis of hierarchy.
...The Egalitarian Left
If we judge leftist ideologies like Marxism by their vision of an ideal society and not by the inevitable results, then we see the most extreme form of an egalitarian society, where nobody is supposed to have any legitimate authority or command over another, even in a voluntarily-organized arrangement. And modern leftists are strongly advocating for a breakdown of any hierarchical system, some even going so far as to call for the abolition of gender.
...The Hierarchical Right
Considering that the right generally opposes everything mentioned earlier that the left supports, that places them in favor of hierarchy. The right is more in favor of established systems that provide structure, and are more in favor of traditional gender roles. Parts of the right (especially the modern right) desire a system of meritocracy and equality of opportunity. Each individual begins where they are, and works to rise higher in the capitalist system. Of course, to rise higher implies that a hierarchy is in place to rise through....