I have just seen the new Dune movie (2021) and I really liked it, but I do not understand the Emperor's actions completely. Note that I have not read any of the books, and I still wish to more-or-less blindly experience the second part of the movie (hoping it gets made) so if the answer hinges on crucial plot points or striking revelations from the second half of the book, I would appreciate a disclaimer before you get to it :-)
The Emperor (I am told he is named Shaddam) is getting the mighty houses Harkonnen and Atreides to go to war and exhaust their resources so he can stay on top himself. To do that he grants Atreides stewardship of Arrakis, and then supports Harkonnen to take it back in a massive planetary invasion, ending the Atreides and rendering the Harkonnen in financial trouble. To support that invasion, he lends the Harkonnen his elite throat-singing soldiers, the Sardaukar.
But as the invasion occurs, on several instances people recognise Sardaukar and understand that the Emperor has violated his neutrality. This is a big deal and enough for Paul to start discussing a "holy war" against the imperial authority, which he implies many houses would join particularly because the Emperor is no longer impartial.
So sending Sardaukar seems like a great misstep on the part of Shaddam. Why did he do that? I understand that in the book these soldiers are disguised as Harkonnen, but I'm sure that they are still recognised. Why would Shaddam not support the Harkonnen in a way that cannot be traced back to him as clearly? E.g., buy them generic mercenaries, or just provide financial means?