As an Ohio State fan this typically comes up at least once a year around bowl season:
"Would you/do you ever root for Michigan?"
I know many people fall in the camp of never rooting for them no matter the circumstance because their hatred for TTUN runs too deep.
I don't think I actively "root" for them ever, but I do hope they win whenever they play certain teams, mostly in bowl games (SEC always, other big teams that stand or have stood in Ohio State's way in the past). While I don't want Michigan to be successful against us and the Big Ten, if they can beat an SEC team or Clemson-type team I'm all for it. It increases the Big Ten's standing in college football, and (recently) it means we've beaten the Michigan team that beat so-and-so.
So, again, I ask: would you/do you ever have a rooting interesting in Michigan?