Although Rafa Benitez will now have managed both Liverpool clubs, he did not go directly between the two.
There are some managers who have directly switched, such as Ron Saunders from Aston Villa to Birmingham City in 1982, before the final few matches of the season (not also forgetting the biggest one the European Cup Final, having done it the proper way by winning the top flight division the previous season). The former owner Doug Ellis had switched from the Bluenoses to Villa a long time ago.
Although both sides were founded on the opposite banks of the Clyde, Ian McCall switched from Clydebank (the 1965-2002 incarnation) to Greenock Morton as Clydebank were a busted flush at the time and played their home matches at Cappielow Park - the home of Greenock Morton.
I do recall there was a Portuguese player who directly switched from Everton to Liverpool over 20 years ago (I think his name began with the letter "X") - unsure whether to avoid Everton losing him on a Bosman.
Kenny Miller went from Hearts to Glasgow Rangers (1872-2012 incarnation), to Wolves, then Celtic, then Derby County, and Glasgow Rangers again. For the Ibrox and Parkhead sides, no player has ever gone directly between the two (obviously to avoid death threats) that I can think of. Another notable player that has played for both sides is Maurice Johnson, after leaving Celtic for MLS in the USA, he returned to Scotland in 1989 declaring that "Celtic are the only team I ever wanted to play for", before controversially accepting the larger offer from David Murray to play for the Ibrox side, therefore upsetting both sets of supporters as they could clearly see that Johnson was only in it for the money. Nobody has seen or heard of him since then after he left Ibrox.