In a number of recent papers the so-called twisted localized mode of the di
screte nonlinear Schrodinger equation has been proposed. Herein, we study t
he existence and stability properties of such modes. We analyze the persist
ence of quasiperiodic modes and study the domains of existence and numerica
l stability of the exact form of such solutions. We identify the bifurcatio
ns through which they lose their stability and follow the behavior of the i
ntrinsic localized modes and their eigenmodes even in the unstable regime.