There is some folksong in the back of my mind with the line, "We both shook hands and parted", in reference to lovers ... if more of the song or the title comes to me, I'll be back ....

As well, it seems to me that there is a southern American type of song in which the speaker proposes that he and the love interest 'shake hands' and go their separate ways, apparently with no hard feelings - again, if I remember more, I'll let you know.