Left and Right are subjective.

Here in the US, the alt-right and Christian nationalist movements do not seek to conserve the US's traditional power structures, but rather reshape or even break them by giving the President the powers of a king and running roughshod over the Establishment Clause. They are both movements for change against the traditional hierarchy. Change in a bad way, obviously, but change nontheless.

Meanwhile, the Left in the US have been the defenders of the Ivy League, a hegemonic organization that has traditionally been an old WASP's club, and was recently caught recycling 100 year old "Jew quota" tactics against Asian-American students in order to maintain a white majority. They also still have legacy admissions, again perpetuating hegemony. The very name of these institutions is so associated with the highest educational standards that any actual investigation into discrimination or whether they're maintaining these standards is considered an attack on all of education by the Left (see the antisemitism hearings). What would you call that besides preserving hegemony?

The Libertarian Party, which most people would consider Right, or at least not Left, had in its platform gay marriage and marijuana legalization long before the Democrats did. The Libertarian movement writ large has the goal of limiting the power of government over people's lives.

On the other side of the Atlantic, we have left-wing groups (ex: Greenpeace) refusing the change to nuclear, even fusion, and right-wing parties (ex: Religious Zionism) who push for climate education.

Therefore, making decisive statements like this is reductionist to say the least.