May I ask how Blavatsky's New Age movement is culturally Christian despite her hating Christianity?
So, being culturally Christian isn't about agreeing with Christianity's beliefs. It's about living within and internalizing elements of Christian culture.
For example, many of our swear words/phrases are related to Christianity - EG, hell, damn, Jesus Fucking Christ, Christ on a cracker, etc.
The word "goodbye" derives from "god be with ye." We often say "bless you" after people sneeze, which comes from "God bless you," which again, has Christian origins.
Christmas being a federal holiday is an example of cultural Christianity. And if you're an atheist celebrating Christmas because you see it as being about family, you're still participating in cultural Christianity.
Now of course, none of these things are inherently bad. In fact, most of cultural Christianity isn't bad. Most of it's pretty neutral. Most of it.
Cultural Christianity also shapes our ideas of what religion looks like, how it functions, and what its purpose is. For example, many western antitheists just assume that all religions want to aggressively spread themselves, all claim to have ultimate truth, and threaten nonbelievers with punishment. Meanwhile, many of these atheists go about their atheism the same way many Christians go about Christianity - treating it as something that needs to be far and wide to save the world and usher in the utopia.
And this brings us to our next point - Christianity shapes how many of us expect the future to unfold. Specifically, a lot of us just sort of think that a utopia is just around the corner (or just imagine that as a thing that can happen if we try hard enough) thanks to Christian millennialism.
Blavatsky's concept of a New Age is basically informed by Christian millennialism. Her whole idea that the spiritually unevolved would be wiped out and a new race of spiritually superior people would take over isn't exactly Christian belief, but it's definitely informed by it.
Now some of you might be thinking, "okay, but Blavatsky drew inspiration from many religions." And you'd be right. But the thing is, she looked at and interpreted these other religions from the perspective of one who was culturally Christian. Additionally, she was taking a perennialist approach to religion, which is a thing Christians have been doing since the early days of Christianity, basically trying to claim that proof of Jesus was found in their own spiritual beliefs and religious traditions. Blavatsky, of course, wasn't looking for proof of Jesus, but she was looking for validation of her own beliefs. Hell, like many Christians before her, she even tries to claim Kabbalah validates her beliefs.
Additionally, she values the Christian Bible as a holy text with spiritual truths that she and everyone should be concerned about. Even if she disagrees with more orthodox interpretations, the fact that she thinks this is a book she needs to concern herself with at all is because of her cultural Christianity. She was informed and influenced by Christian modes of occultism and esotericism.
Ultimately, being culturally Christian has nothing to do with whether you embrace or even like Christianity's spiritual doctrines. It's about living in a Christianized society and conforming to any of its Christianity-derived assumptions, mores, and customs, regardless of what they have to do with any official church doctrine.