Update: after making it through volume 11, I'm revising my opinion upwards. I originally gave In/Spectre a 9/9/8.5, for an overall rating of 9. The substance of my original review (below) still accurately describes my impressions, but I'm increasingly impressed by the quality of the writing, the intricacy and soundness of the mysteries, and the characterization of the main protagonist. The mysteries in In/Spectre are unlike anything you'll encounter in any other manga—or, for that matter, in any novel, film, or show. They're genuinely excellent (and I say this as a huge fan of mysteries). And Kotoko is totally unlike any other fictional detective, but she's the most compelling one I've ever read. The fantastic elements of In/Spectre somehow add to her plausibility as a character.
At this point, my only criticism is that I would like for Kuro's character to be explored in the same depth as Kotoko's. But I feel confident that's coming (and may even already have happened in the 4 remaining English volumes that have currently been released).
Original Review:
This is a really neat series. As far as I know, its premise is unique in any medium, not just in manga. The fundamental concept is a series of mysteries. (So far, there is one long arc that spans several volumes, and there are eight or so smaller mysteries of about a half-volume each.) But it departs from the ordinary mystery "formula" in two ways.
First, the protagonist's goal is rarely to uncover the truth. She usually knows the truth already, because she can talk to ghosts and demons, who are always happy to tell her what they know. Instead her goal is to produce a satisfying story. For instance, a snake god asked the protagonist to find out why a murderer had disposed of her victim's body in the snake god's pond. The protagonist suggests multiple explanations, each of which the snake god rejects as irrational in some respect, until she hits on a convoluted story that the god finds compelling. The snake god accepts her answer and she departs with his gratitude. But it turns out that the truth was one of the simpler stories that the snake god had rejected, and the protagonist knew this—she was just trying to find a lie that would satisfy him.
I like this a lot. One of the problems with mysteries, in my opinion, is that the detective's reasoning is too fine; all it takes is the slightest irrationality on the part of a criminal, or the slightest error of judgment on the detective's part, and the solution falls apart. But when the "detective" is instead trying to compose a satisfying story, the same mental processes and logical puzzles are involved as in a traditional mystery, but at the end the reader is not left with the feeling that the solution was contrived by the author.
Second, there is another supernatural component. The protagonist's "boyfriend" (it's complicated) heals each time he would die, and he can manipulate the future a bit each time he "dies." So often the protagonist's goal is to solve some problem by making a solution possible, while her boyfriend uses his powers to make the solution effective.
I like this a lot, too. First, there is a nice paranormal creepiness associated with his powers, and that adds to the flavor of the manga. Second, and more importantly, his power of manipulating the future also contributes to the sense that the protagonist's planned solutions are reasonable and not overly contrived.
All in all, this is a great series and I would recommend it to anyone who likes mystery novels. My only complaint is that I would have liked the romance to be slightly more prominent. It's less central than you would expect from the series description.