Welcome to the world of phase cancellation
hint: you got more control over it by doin this in your DAW. Split the stereo signal into 2 mono signals (left and right) on 2 tracks of your DAW pan both to the mid and reverse the phase on ONE of the tracks (doesn't matter which one) with one of the following methods:
1. phase reverse switch on the track (if your DAW provide such a button)
or
2. VST phase reverse plugin on one of the tracks insert
or
3. Mark the waveform on one track and search the DAW for something like edit -> invert
if you've done it you can more or less (depend on the song) control a lil bit what you will hear/not hear by increase or decrease the volume on one of the tracks.
HEY! NOT THAT MUCH SON! just in 0.1dB steps

Your experience with the cable is something I know too. It was the first time I heard that effect and it happened accidently. It was 1998, I've had a portable CD player connected to my stereo and listened to "More bounce to the ounce" from Zapp. but something was strange. Most of the music was gone and it was nearly an acapella. WTF? Later I've noticed that the ground connection of the cable was broken and this is what caused this effect. It was really enlighten me cause I started listening to so many songs and realized all the details in some instruments that are placed in the back of the songs.
I took a switch from a bedside lamp and modified my headphones. The switch cut the groundcord and there was the effect even when I was on the way..........nice memorys of that time........
peace
Listen to (and please do yaself a favor and use good copies of this songs cause with this technique you can really hear howbad mp3 really sound)
Michael Jackson - Billy Jean (do think twice, do think twice ahooo

)
A tribe called quest - Award tour (dodat dodat dodo dat dat dat , come on do dat do dat doo doo dat dat dat, ok do dat do dat do do dat dat dat)
Zapp - More bounce