Magnetohydrodynamic (MHD) waves are ubiquitous in the solar atmosphere. Alfvén waves and magneto-sonic waves are particular classes of MHD waves. These wave modes are clearly different and have pure properties in uniform plasmas of infinite extent only. Due to plasma non-uniformity, MHD waves have mixed properties and cannot be classified as pure Alfvén or magneto-sonic waves. However, vorticity is a quantity unequivocally related to Alfvén waves as compression is for magneto-sonic waves. Here, we investigate MHD waves superimposed on a one-dimensional non-uniform straight cylinder with constant magnetic field. For a piecewise constant density profile, we find that the fundamental radial modes of the non-axisymmetric waves have the same properties as surface Alfvén waves at a true discontinuity in density. Contrary to the classic Alfvén waves in a uniform plasma of infinite extent, vorticity is zero everywhere except at the cylinder boundary. If the discontinuity in density is replaced with a continuous variation of density, vorticity is spread out over the whole interval with non-uniform density. The fundamental radial modes of the non-axisymmetric waves do not need compression to exist unlike the radial overtones. In thin magnetic cylinders, the fundamental radial modes of the non-axisymmetric waves with phase velocities between the internal and the external Alfvén velocities can be considered as surface Alfvén waves. On the contrary, the radial overtones can be related to fast-like magneto-sonic modes.