And there is a history to The Game, which animates the rivalry every year (see
Michigan-Ohio State rivalry - Wikipedia, the free encyclopedia).
I can't find any reference to an LSU-Auburn rivalry anywhere, in a way that would be considered like The Game. Certainly, Auburn fans consider Alabama their rival, not LSU.
For instance, when ESPN asked viewers to vote on the top college football rivalry of all time, Auburn-Alabama made it into the final four, but Ohio State-Michigan was voted #1
(http://espn.go.com/page2/s/bracket/collegefootball/rivalry.html). LSU did not feature in any rivalries in either poll.
Earlier, in another ESPN poll, fans voted the Ohio State-Michigan rivalry as the top rivalry in sports in another ESPN poll. As they said,
1. Michigan vs. Ohio State
When Ohio Stadium opened in 1922, Michigan spoiled the party with a 22-0 victory. The rivalry was heated in the early days as both have been long-time college football powers. But it got even hotter in 1969, when Bo Schembechler took over as Michigan's coach and upset Woody Hayes' No. 1-ranked, undefeated Buckeyes. Four times in the next six years, both teams were ranked in the top five when they met. In 1970 and 1973, both were undefeated (they tied 10-10 in '73). From 1970 through 1975, Michigan entered without a loss every year. The Wolverines won just once. Ohio State was 9-0-1 in 1993, 11-0 in 1995 and 10-0 in 1996. The Buckeyes lost each time. That
is rivalry. ESPN.com - ENDOFCENTURY - ESPN.com's 10 greatest rivalries.
Another Buckeye's rivalry ranked #5 (Jack Nicklaus vs Arnold Palmer).
For goodness sakes, there was even an HBO program dedicated to The Game this year.
While we respect the LSU Tigers football team that Ohio State is about to play, to suggest even tangentially, that LSU has anything like the football tradition of Ohio State, or a rivalry of the type between Ohio State and TSUN, is something that you would have very great difficulty getting anyone outside your fanbase (and inside your fanbase) to agree.