  We identify a relationship between the solutions of a nonsymmetric algebraic
T-Riccati equation (T-NARE) and the deflating subspaces of a palindromic matrix
pencil, obtained by arranging the coefficients of the T-NARE. The interplay
between T-NARE and palindromic pencils allows one to derive both theoretical
properties of the solutions of the equation, and new methods for its numerical
solution. In particular, we propose methods based on the (palindromic) QZ
algorithm and the doubling algorithm, whose effectiveness is demonstrated by
several numerical tests
