O'Connor posted an open letter to Cyrus on Thursday after Cyrus told journalists she had modeled her "Wrecking Ball" video on O'Connor's famous "Nothing Compares 2 U."
She warns the former "Hannah Montana" star that it would be self-defeating to market her sexuality.
"I am extremely concerned for you that those around you have led you to believe, or encouraged you in your own belief, that it is in any way `cool' to be naked and licking sledgehammers in your videos," O'Connor writes in reference to the sexually charged "Wrecking Ball" video.
She also cautions the 20-year-old Cyrus that the music business will exploit her and then abandon her.
The advice didn't go down well.
Cyrus responded on Twitter by mentioning O'Connor's struggles with bipolar disorder.
In a reference to troubled actress Amanda Bynes, who has been treated in a psychiatric hospital in California, Cyrus tweeted: "Before Amanda Bynes...There was..." and posted O'Connor's own tweets about seeking psychiatric help.
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