I'm very glad you mentioned this. Cheney was the House Minority Whip after Trent Lott won his U.S. Senate seat in Mississippi. This made Cheney second in command among House Republicans to Illinois Rep. Robert Michel, a moderate, and also the leading conservative. Gingrich only really distinguished himself among the House Republicans when he ran to succeed Cheney as Minority Whip after Cheney was appointed Secretary of Defense following the Tower nomination's failure. The George H. W. Bush White House made a spirited effort to support a moderate against Gingrich, whom they thought was a demagogue (where did they get that idea, I wonder?). Gingrich won despite their efforts, which made him a hero in the House Republican caucus. The moderate Republicans weakened further in the 1990 and 1992 elections. Thus, when Michel announced his retirement in 1994, Gingrich was the obvious replacement as head of the party. Of course at the same time as Gingrich went to the head of the line of the Republicans, the Republicans defeated the Democrats for control of the House in their legendary 1994 victory. So Gingrich went straight from being the Minority Whip in one Congress to being Speaker the next. And he lasted in the role only four years, his rival Tom DeLay having much greater skill in wielding power in the U.S. House, partly by never actually holding the Speaker's chair himself.
So now, if Tower is confirmed as U.S. Secretary of Defense, Cheney stays as House Republican Minority Whip, and there is no spirited race to fill the position. This means Cheney becomes the obvious leader of the House Republicans upon Michel's retirement, and given the very high likelihood of a House Republican victory in 1994 under either Gingrich or Cheney's leadership, Cheney quite probably becomes Speaker of the U.S. House in 1994.
This could actually be terrible for Democrats, because though he became very tone deaf with respect to public opinion in his last years in office, Cheney was a very skilled politician, and certainly a better strategist than Gingrich. I think Speaker Cheney means the rest of Clinton's term would become much tougher, and that itself could have significant secondary effects