Just finished watching the installment of the HBO John Adams series where both Washington and Adams are inaugurated. Two words: Dork Heaven. Not sure about the accuracy of the portrayal (haven't gotten to that part in the McCullough book yet), but in the series Washington is sworn in before masses of cheering patriots. Adams is sworn in before Congress, and gets his most rousing applause during his inaugural address when he praises Washington.
I read in a New Yorker article that Adams' inaugural address was both "indefinite" (a large percentage of the words in the address were "if") and "cumbrous" (the third to last sentence was over 700 words long).
Meanwhile, if that's not enough inaugural trivia for you, check out these "Quick 10 Facts" about Presidential inaugurations from Mental Floss. My favorite: Andrew Johnson was so drunk he couldn't speak clearly during his inauguration.