For some reason, I want this to be a really good blurb. I'm going to pick apart each piece and hopefully the feedback will help. This is just my opinion on your blurb and I am not the end all, be all, authority on blurb writing.
Cattleya Hoskin is a magazine reporter on permanent assignment to chronicle the casework of Pittsburgh PI, Lupa Schwartz.
This introduces two characters in one sentence. I still don't know which of these two is the main character after reading the rest of the blurb. If they are both main characters then you need to just pick one. Which one of these characters changes and grows through the story? Which one's actions and choices are what drives the plot and conflict resolution? We need a clear main character and plot. I also find this introduction a little weird. A magazine reporter is permanently assigned to report on a private investigator? This doesn't make any sense and it doesn't contribute to the blurb. This relationship is false detail for the rest of the blurb. They could simply be friends, sisters, or strangers and the rest of the blurb is unaffected.
She returns home after an extended hospital stay following a nearly fatal attack, only to find that her ex-boyfriend, Ulric, has gone missing.
Another false detail sentence. If this sentence is removed, the rest of the blurb is unaffected. The only thing this sentence does is tell us Ulric is her boyfriend. The attack and that he's missing, does not effect the blurb in any way. It's an independent piece of information that the blurb does not use to build upon.
Little by little she learns that Schwartz and Ulric have been covertly unraveling a centuries-old scheme, and the conspirators have driven Ulric underground.
This is the first hint we get of a primary conflict. Unraveling a centuries-old scheme. This is good. The rest of the sentence doesn't help it out. First is how did she miss this if she's a reporter permanently assigned to one of the investigators? Next we have the use of "driven underground" This choice of words doesn't fit well with "has gone missing" with the earlier sentence. If you want to use this, they should be aligned better.
First paragraph overall: I would get rid of the boyfriend. I would introduce the main character with some sort of unique qualifier for them to be the main character. I would introduce the PI and the conflict. This would give you character, character, conflict. This format works well with blurbs. Right now you have character, character, conflict, character, conflict, conflict. It's just way too much.
Complicating matters, a mystery woman has come seeking Schwartz's help.
I don't know on this one. Mostly because I think the main conflict is the centuries old scheme, and she is a part of that conflict, but I can't be sure. It is generic though and uninteresting. Also a whole two sentences later the mystery is revealed so the overall value of this sentence is approaching null.
Though still very much alive, she wants him to expose her murderer after the fact.
This is a confusing sentence. I know what you're trying to say, but it's not working here with the clever linear reversal. This is also a spoiler. We know she's going to die. There are going to be several other ways to deliver this setup to the reader that is going to flow better. Give us something interesting about this woman who hires the PI then shows up murdered.
When she reveals herself to be a member of the very family Ulric has been investigating, Schwartz and Cattleya can't help but become involved.
Spoiler to the first sentence of this paragraph.
Second paragraph overall: We could get a real good look at the conflict and plot here, but it really falls short with the spoiler and the clever, hire you to investigate my own murder before I die part. Again I would ditch the boyfriend. We don't need more characters and his plight is just frosting the same primary conflict.
The placement of the world's tall towers, the esoteric secrets of sacred geometry, a genetic line that stretches back to long before Charlemagne, and an ancient shadow-government all coalesce to drive Schwartz and Cattleya offgrid in an attempt to unravel the deepest mystery of their lives.
You switch from a third person limited perspective to a narrator here. Instead of building tension and excitement, the blurb just tells us what's going on. All the previous details have lost most of their value to keep a reader engaged. You're summarizing the blurb.
If they succeed, they just might topple an empire. If they fail, there may be no place for them in the New World Order -- or anywhere else.
I already talked about how this is a spoiler or a red herring lie. I don't see a way for this sentence to do anything other than to get a customer to look for the answer in the reviews. You want the customer to be intrigued to where they want to buy the book. I feel that this last bit can only hurt the success of your book, not help it in any way.
3rd paragraph: I would nuke it. There is nothing here worth saving. It's a narration that brings in a lot of weird details. All of these details should have been sprinkled in throughout the blurb. You shouldn't have to tell us that it's a secret society thriller/adventure story. The blurb should be able to do that on it's own. What magazine does she work for? Is it one that would drag her into a world like this? If so that could be your interesting detail that shows us what she's getting involved in. Drop us right into the action with this woman who may be descended from Charlemagne. Instead we are hit over the head with tall towers, math, history and governments.
I can sense that there is a real gem of a story here, but I just can't find it because it's surrounded by overburden. It just needs to be refined and all the muck and silt washed away. The topics and genera you have chosen to write in are very fun to read and their movie and tv counterparts are fun to watch. I have little doubt your story is as good if not better than they are.