It sounds strange to me as well. Has the loser always attended the inauguration?
The answer is no.
McGovern did not attend Nixon’s in 1973:
http://www.nytimes.com/1973/01/14/archives/mcgovern-will-miss-inauguration-of-nixon.html
It looks like Mondale was a no-show in 1985:
http://articles.latimes.com/1985-01-17/news/mn-7700_1_mondale
Dukakis didn’t show in 1989:
http://www.nytimes.com/1989/01/21/us/the-41st-president-a-disappointed-dukakis-wishes-bush-well.html
Romney did not attend in 2013:
http://www.nydailynews.com/news/politics/mitt-romney-president-obama-inauguration-article-1.1244471
The article is clumsily written. Mondale, McGovern, Romney, Dukakis were not ex-Presidents. But Bill Clinton being an ex-President, would attend the nomination. Hillary did not want him to go alone...