A linear composition of a positive integer N is a list of positive integers (called parts) whose sum equals N. We distinguish two kinds of cyclic compositions, which we call C-type and CR-type. A CR-type cyclic composition of N is an equivalence class of all linear compositions of N that can be obtained from each other by a cyclic shift, while a dihedral composition is an equivalence class of all linear compositions of N that can be obtained from each other by a cyclic shift or a reversal of order. A linear Carlitz composition is one where adjacent parts are distinct. A C-type cyclic Carlitz composition is a linear Carlitz composition whose first and last parts are distinct, whereas a CR-type cyclic Carlitz composition is an equivalence class of C-type Carlitz compositions that can be obtained from each other by a cyclic shift. We distinguish two kinds of linear palindromic compositions (type I and type II). We derive generating functions for the number of type II linear palindromic Carlitz compositions, and we provide a new proof of a result by J. Taylor about C-type Carlitz compositions. Using these results, we derive formulas about CR-type Carlitz compositions, symmetrical CR-type compositions, and dihedral Carlitz compositions.