In fact, other than Romney and Dole, every election loser since GHWB lost has had a position where the norm was that they would have had to sit publicly, in the stands, as the victor was sworn in. Al Gore, in addition, in the role of VP, had to officiate over the count where Bush was given Florida, which Gore, it turns out actually, provably (though a full audit of the whole state done by newspapers was not completed until 9/11/01) did win. Then I remember that he sat with Clinton as Bush took the oath of office.
With GHWB, I remember the tv coverage of the Clintons going to the White House and having the typical formal meeting of the incoming family and the outgoing one -- in this case, the man who lost the election. I know that GHWB was at the ceremony when Clinton was sworn in, but I actually do not remember anything about that.
I remember Kerry, classy and dignified as always, sitting with fellow Senators on what should have been his day. One positive was watching as the press panned to Kerry often before the ceremony - that many of his peers hugged or put their arms around his shoulder. It was tough to watch. I remember John McCain less, but I do remember him sitting with that same group of peers -- and they also understood how hard that had to be for him.
Like everyone of these men who lost, who honored the tradition of US elections and inaugurations, I am sure that HRC will be a gracious dignified presence publicly ... and we all know that she, like those men, will be intensely sad.
One other note, though 2009 was a Democratic triumph, this really is the SECOND time that HRC had to sit at an inauguration that she had hoped, prayed and worked to make hers.