My exposure to James P. Hogan has been limited. I liked his Gentle Giants of Ganymede trilogy, and I thought Two Faces of Tomorrow was okay. I thought I'd expand my reading a bit with The Multiplex Man. Either it's not as good as the other books, or I simply don't enjoy Hogan as much as the younger me.
Hogan's world-building premise is a simple inversion, with the West as increasingly authoritarian and directive, while the former Soviet states have become free and laissez faire. There are sizable infodumps, but it's a mildly interesting setup. Unfortunately, Hogan focuses so heavily on a libertarian message that it gets in the way of the story. Heinlein did this more smoothly, while Hogan appears to suggest that any regulation at all is a freedom-killer. In some ways it is; the whole point of rules is to limit behaviour to agreed norms. But Hogan never really suggests how his desired anarchic society would function practically; life was just better when we could freely smoke, pollute, not pay taxes, and do other fun, free stuff. It's reminiscent of Heinlein's 'citizen=armed male' formula, but without the conviction.
Although the book was written in the 1990s, it seems to carry the ballast of long-gone decades. Most of the cast are men. The text includes the line "with another soldier who gave his name as Schott, and a black called Lowe" - who is also a soldier. There's not a lot more of this, but it doesn't sit well. Probably the kind of PC nit-picking that freedom-loving libertarians despise.
The story itself, about transposition of identities and skills across bodies, is fairly well constructed, with distinct personalities among the various identities. There are portions that are weak on credibility, and there's a certain amount of 'happy coincidence'. A phrase late in the book encapsulates the problem - "All of a sudden, something inside him didn't feel right." There just isn't enough buildup for some of the key changes. Hogan does do a nice job of presenting the confusion of the key actors, and a fair job of sequencing sections of the plot in an interesting way, though Hogan fudges with some 'no one will ever know'. The climax, however, is visible a long way off, and isn't saved by satisfying final resolution. Though this is definitely soft SF, there's a little too much hand-waving about the powers of the story's science, which achieves a little more than it should.
All in all, a workmanlike SF thriller in an interesting if tired environment. It's not terrible, but there's nothing that suggests you should go out of your way to buy the book, or even to shift it higher in your reading stack. If you want to read Hogan, I suggest his Giants series instead. If you want libertarian escapism, read Heinlein.