<p>We study time-modulated Taylor–Couette flow of purely polymeric liquids for the case in which the inner and outer cylinder’s angular velocities counter-oscillate around zero mean at given amplitude and frequency. When the rheological behavior of these liquids is well described by the Upper Convected Maxwell model a linear stability analysis is performed and discussed based on Floquet theory to shed light on system transitions from laminar to Taylor–vortex flows. While the Newtonian version of this flow configuration is characterized only by two synchronous bifurcations, it is shown that the fluid elasticity leads to the appearance of numerous new synchronous and Neimark–Sacker bifurcations. Owing to strong competition between these instability modes, several codimension-two bifurcation points are detected featuring stability diagrams with several branches, discontinuities and cusp points. Moreover, it turns out that the fluid elasticity modifies the flow reversal properties of the Taylor vortices in the sense that a new reversing flow with cessation, not observed in the Newtonian case, is detected at a specific frequency range. In addition and by gradually increasing the fluid elasticity, a strong destabilizing effect in the low frequency limit is observed in contrast to the moderate and high frequency limits where either a stabilizing or a destabilizing effects are detected.</p>