The lyrics of Counterparts continue the trends of Roll the Bones with dark and emotional themes being the primary focus. Some songs are heavy-sounding tracks like "Animate" and "Stick it Out", which topped the BillboardAlbum Rock Tracks chart for four weeks in late 1993, becoming the band's fifth single to do so. "Leave That Thing Alone" earned a Grammy nomination for "Best Instrumental". It has been reissued and remastered twice: once in 2004 as a continuation of "The Rush Remasters" series and again in 2013 as a part of the box set The Studio Albums 1989-2007.
The albums front cover contains some innuendo: A Nut and Bolt being directed towards each other, similar to how sexual interaction works between a male and female. In an interview, Neil Peart explained how he viewed the shaping of the "Modern Man" of the 1980's was to be that of the opposite gender, in which he was very controversial about. He further uses these terms in the track "Animate" of how the roles of being both strong and sensitive should be equalized. This may have led to the possible generation of the artwork for the cover.