I came across this post on Listverse titled "10 Pieces of Music to Make Your Ears Bleed" in which the author lists 10 "unlistenable" pieces of classical music, almost all of them bedrock staples of the 20th century repertoire (‘Gesang Der Junglinge’ by Stockhausen, Penderecki's Threnody, Berg's Wozzeck, Varèse's Ionisation) and highly influential on later generations.
The post brought to mind the quote from Charles Ives - "Is not beauty in music too often confused with something which lets the ears lie back in an easy-chair?" [from the "Postface to 114 Songs" which can be read here.]
Rather than describing this music as "unlistenable", I would say this is music to which you have to listen, it requires you to be an active listener, to be a participant in the music making experience, not just sit there and passively give up a predetermined response.
Of course the author does not show much knowledge about what she presents; Schoenberg is described as believing "music no longer needs melody, harmony, or any of those other pesky reasons for which most of us listen to it", rather than believing that melody and harmony need not be confined by the rules which had governed them for the past few centuries. The introduction presents the author's premise that "classical music is often thought of as refined, relaxing—and, well, classy. But there’s a dark side to classical..." There is much by which to be bothered in this sentiment. Mainly, the offense taken that a falsely conceived perception is not true. It's as if someone who thinks Shakespeare is all mushy love poetry recited by men in tights ends up feeling betrayed by the violence and unseemly behavior of Macbeth or Othello.
Classical music offers a large array of expression and emotion largely because it was (and is) written by human beings, creatures known to possess a large array of expression and emotion.
I've not researched further to see what music the author actually likes, but I heartily recommend checking out all the works listed in her article, because I respect my listeners (and readers) as people who enjoy a challenge, and don't like to settle for the usual blah.