We generalize recent work of Andrews, Just, and Simay on modular palindromic compositions and anti-palindromic compositions by viewing all compositions partially (modular) palindromic or anti-palindromic. More precisely, we enumerate compositions by the extent to which they are (modular) palindromic or anti-palindromic. We obtain various closed formulas from generating functions and provide bijective proofs for many of them. We recover some known results of Andrews, Just, and Simay and discover new connections with numerous sequences in The On-Line Encyclopedia of Integer Sequences (OEIS).