Alternans, a beat-to-beat temporal alternation in the sequence of heartbeats, is a known precursor of the development of cardiac fibrillation, leading to sudden cardiac death. The equally important precursor of cardiac arrhythmias is the rotating spiral wave of electro-mechanical activity, or reentry, on the heart tissue. Here, we show that these two seemingly different phenomena can have a remarkable relationship. In well controlled in vitro tissue cultures, isotropic populations of rat ventricular myocytes sustaining a temporal rhythm of alternans can support period-2 oscillatory reentries and vice versa. These reentries bear "line defects" across which the phase of local excitation slips rather abruptly by 2pi, when a full period-2 cycle of alternans completes in 4pi. In other words, the cells belonging to the line defects are period-1 oscillatory, whereas all of the others in the bulk medium are period-2 oscillatory. We also find that a slowly rotating line defect results in a quasi-periodic like oscillation in the bulk medium. Some key features of these phenomena can be well reproduced in computer simulations of a nonlinear reaction-diffusion model.