LSQR and CGLS represent well known Krylov subspace methods for the solution of linear approximation problems including ill-posed ones. In order to accelerate convergence or impose solution constraints, various preconditioning strategies have been proposed. Basic LSQR and CGLS are mathematically equivalent, but relations among their preconditioned variants have been studied only partially. In this paper, we do not restrict to particular applications, but we algebraically describe underlying subspaces, optimality conditions, and relate quantities generated by selected preconditioned variants. Furthermore, motivated by available right preconditioned CGLS avoiding transformation of coordinates, we give a transformation free right preconditioned LSQR algorithm.