Extended Schur functions and bases related by involutions

We introduce two new bases of QSym, the flipped extended Schur functions and the backward extended Schur functions, as well as their duals in NSym, the flipped shin functions and the backward shin functions. These bases are the images of the extended Schur basis and shin basis under the involutions $\rho$ and $\omega$ on the quasisymmetric and noncommutative symmetric functions, which generalize the classical involution $\omega$ on the symmetric functions. In addition, we prove a Jacobi-Trudi rule for certain shin functions using creation operators. We define skew extended Schur functions and skew-II extended Schur functions based on left and right actions of NSym and QSym respectively. We then use the involutions $\rho$ and $\omega$ to translate these and other known results to our flipped and backward bases.