Freddy’s “It depends on the President and Vice President” answer is very much on target, though I’d have to quibble with a few details.
In 1932 Garner threw his support behind Roosevelt in exchange for the Vice Presidency. They were, for the next 4+ years, respectful of each other as disparate parts of the Democratic coalition. Garner had an inside track for expressing the views of Southern Democrats of the older. more conservative school on New Deal legislation, and Roosevelt listened to it and used it to help shape how he got New Deal bills through Congress. If, for example, Garner reported that “Sens. X, Y, and Z are reliable votes for your farm bill, but you’ve got to let them make speeches on the Senate floor decrying it, for consumption by the home folks, first,” that was useful information on how to put together a majority for that bill, and not to be deceived by their seeming opposition. If Garner said, “You need to take out Section 7 of this bill – it’ll never sell in the South with that in. Pull it, and our boys can argue that the rest is needed, and that they got that concession from you,” FDR would probably pull Section 7. The Court-packing plan was what precipitated the break between them, and most of Garner’s second term was more or less in opposition – but the role he played in his first term and the beginning of the second should not be overlooked.
Ike, a consummate politician, knew how to play the public. He himself didn’t take controversial stands, in general (enforcing the SCOTUS integration decisions being one major exception, and that out of his respect for government by law). If a controversial proposal needed to be vetted, he had Nixon trot it out and take the flak if it did prove controversial. If maverick Republicans needed to be brought into line or publicly spanked, Ike, who traded on his pater patriae image to forge a national consensus, would not descend to the level of party leader; Nixon did the dirty work, with Ike’s full consent and encouragement.
Nixon himself tried to use Agnew in a similar role when he finally became President himself, but the country was too divided over Vietnam and related issues at that point, and Nixon himself was too much a partisan, controversial figure to play the Ike role.
Rockefeller more or less reprised the Garner role, speaking for the liberal Republicans, during Ford’s Presidency, but of course there were few instances in those two years for him to do so. But Ford, a staunch conservative as a Congressman, was smart enough to balance his Administration by nominating Rockefeller, the leading liberal in the GOP, in an effort to reconstruct a national coalition/consensus.